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A001107
10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).
(Formerly M4690)
+10
131
0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers on the negative y-axis (see Example section).
Number of divisors of 48^(n-1) for n > 0. - J. Lowell, Aug 30 2008
a(n) is the Wiener index of the graph obtained by connecting two copies of the complete graph K_n by an edge (for n = 3, approximately: |>-<|). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. - Emeric Deutsch, Sep 20 2010
This sequence does not contain any squares other than 0 and 1. See A188896. - T. D. Noe, Apr 13 2011
For n > 0: right edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012
Sequence found by reading the line from 0, in the direction 0, 10, ... and the parallel line from 1, in the direction 1, 27, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Jul 18 2012
Partial sums give A007585. - Omar E. Pol, Jan 15 2013
This is also a star pentagonal number: a(n) = A000326(n) + 5*A000217(n-1). - Luciano Ancora, Mar 28 2015
Also the number of undirected paths in the n-sunlet graph. - Eric W. Weisstein, Sep 07 2017
After 0, a(n) is the sum of 2*n consecutive integers starting from n-1. - Bruno Berselli, Jan 16 2018
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.3 p. 33.
Emilio Apricena, A version of the Ulam spiral.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 344.
Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Leo Tavares, Illustration: Conjoined Hexagon/Square Pairs
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Decagonal Number.
Eric Weisstein's World of Mathematics, Graph Path.
Eric Weisstein's World of Mathematics, Sunlet Graph.
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = A033954(-n) = A074377(2*n-1).
a(n) = n + 8*A000217(n-1). - Floor van Lamoen, Oct 14 2005
G.f.: x*(1 + 7*x)/(1 - x)^3.
Partial sums of odd numbers 1 mod 8, i.e., 1, 1 + 9, 1 + 9 + 17, ... . - Jon Perry, Dec 18 2004
1^3 + 3^3*(n-1)/(n+1) + 5^3*((n-1)*(n-2))/((n+1)*(n+2)) + 7^3*((n-1)*(n-2)*(n-3))/((n+1)*(n+2)*(n+3)) + ... = n*(4*n-3) [Ramanujan]. - Neven Juric, Apr 15 2008
Starting (1, 10, 27, 52, ...), this is the binomial transform of [1, 9, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=1, a(2)=10. - Jaume Oliver Lafont, Dec 02 2008
a(n) = 8*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Jul 10 2010
a(n) = 8 + 2*a(n-1) - a(n-2). - Ant King, Sep 04 2011
a(n) = A118729(8*n). - Philippe Deléham, Mar 26 2013
a(8*a(n) + 29*n+1) = a(8*a(n) + 29*n) + a(8*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n >= 1} 1/a(n) = Pi/6 + log(2) = 1.216745956158244182494339352... = A244647. - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Aug 28 2016: (Start)
E.g.f.: x*(1 + 4*x)*exp(x).
Sum_{n >= 1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 2*log(2) + 2*sqrt(2)*log(1 + sqrt(2)))/6 = 0.92491492293323294695... (End)
a(n) = A000217(3*n-2) - A000217(n-2). In general, if P(k,n) be the n-th k-gonal number and T(n) be the n-th triangular number, A000217(n), then P(T(k),n) = T((k-1)*n - (k-2)) - T(k-3)*T(n-2). - Charlie Marion, Sep 01 2020
Product_{n>=2} (1 - 1/a(n)) = 4/5. - Amiram Eldar, Jan 21 2021
a(n) = A003215(n-1) + A000290(n) - 1. - Leo Tavares, Jul 23 2022
EXAMPLE
On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie along the negative y-axis, as seen in the example below:
99 64--65--66--67--68--69--70--71--72
| | |
98 63 36--37--38--39--40--41--42 73
| | | | |
97 62 35 16--17--18--19--20 43 74
| | | | | | |
96 61 34 15 4---5---6 21 44 75
| | | | | | | | |
95 60 33 14 3 *0* 7 22 45 76
| | | | | | | | | |
94 59 32 13 2--*1* 8 23 46 77
| | | | | | | |
93 58 31 12--11-*10*--9 24 47 78
| | | | | |
92 57 30--29--28-*27*-26--25 48 79
| | | |
91 56--55--54--53-*52*-51--50--49 80
| |
90--89--88--87--86-*85*-84--83--82--81
[Edited by Jon E. Schoenfield, Jan 02 2017]
MAPLE
A001107:=-(1+7*z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* Harvey P. Dale, May 08 2012 *)
Table[PolygonalNumber[RegularPolygon[10], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
Table[4 n^2 - 3 n, {n, 0, 49}] (* Alonso del Arte, Jan 24 2017 *)
PolygonalNumber[10, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 27}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)
PROG
(PARI) a(n)=4*n^2-3*n
(Magma) [4*n^2-3*n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 05 2014
(Python) a=lambda n: 4*n**2-3*n # Indranil Ghosh, Jan 01 2017
def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.
x, y = 1, 1
yield 0
while True:
yield x
x, y = x + y + 8, y + 8
A001107 = aList()
print([next(A001107) for i in range(49)]) # Peter Luschny, Aug 04 2019
CROSSREFS
Cf. A007585, A028994.
Cf. A093565 ((8, 1) Pascal, column m = 2). Partial sums of A017077.
Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003215.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A002939
a(n) = 2*n*(2*n-1).
+10
90
0, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Write 0,1,2,... in a spiral; sequence gives numbers on one of 4 diagonals (see Example section).
For n>1 this is the Engel expansion of cosh(1), A118239. - Benoit Cloitre, Mar 03 2002
a(n) = A125199(n,n) for n>0. - Reinhard Zumkeller, Nov 24 2006
Central terms of the triangle in A195437: a(n+1) = A195437(2*n,n). - Reinhard Zumkeller, Nov 23 2011
For n>2, the terms represent the sums of those primitive Pythagorean triples with hypotenuse (H) one unit longer than the longest side (L), or H = L + 1. - Richard R. Forberg, Jun 09 2015
For n>1, a(n) is the perimeter of a Pythagorean triangle with an odd leg 2*n-1. - Agola Kisira Odero, Apr 26 2016
From Rigoberto Florez, Nov 07 2020 : (Start)
A338109(n)/a(n+1) is the Kirchhoff index of the join of the disjoint union of two complete graphs on n vertices with the empty graph on n+1 vertices.
Equivalently, the graph can be described as the graph on 3*n + 1 vertices with labels 0..3*n and with i and j adjacent iff iff i+j> 0 mod 3.
A338588(n)/a(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.
Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j> 0 mod 3.
These graphs are cographs. (End)
a(n), n>=1, is the number of paths of minimum length (length=2) from the origin to the cross polytope of size 2 in Z^n (column 2 in A371064). - Shel Kaphan, Mar 09 2024
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
Eric Weisstein's World of Mathematics, Kirchhoff Index
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
Sum_{n >= 1} 1/a(n) = log(2) (cf. Tijdeman).
Log(2) = Sum_{n >= 1} ((1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ...) = Sum_{n >= 0} (-1)^n/(n+1). Log(2) = Integral_{x=0..1} 1/(1+x) dx. - Gary W. Adamson, Jun 22 2003
a(n) = A000384(n)*2. - Omar E. Pol, May 14 2008
From R. J. Mathar, Apr 23 2009: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*(1+3*x)/(1-x)^3. (End)
a(n) = a(n-1) + 8*n - 6 (with a(0)=0). - Vincenzo Librandi, Nov 12 2010
a(n) = A118729(8n+1). - Philippe Deléham, Mar 26 2013
Product_{k=1..n} a(k) = (2n)! = A010050(n). - Tony Foster III, Sep 06 2015
E.g.f.: 2*x*(1 + 2*x)*exp(x). - Ilya Gutkovskiy, Apr 29 2016
a(n) = A002943(-n) for all n in Z. - Michael Somos, Jan 28 2017
0 = 12 + a(n)*(-8 + a(n) - 2*a(n+1)) + a(n+1)*(-8 + a(n+1)) for all n in Z. - Michael Somos, Jan 28 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 - log(2)/2. - Amiram Eldar, Jul 31 2020
EXAMPLE
G.f. = 2*x + 12*x^2 + 30*x^3 + 56*x^4 + 90*x^5 + 132*x^6 + 182*x^7 + 240*x^8 + ...
On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step in any of the four cardinal directions and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie along one of the diagonals, as seen in the example below:
.
99 64--65--66--67--68--69--70--71--72
| | |
98 63 36--37--38--39--40--41--42 73
| | | | |
97 62 35 16--17--18--19--20 43 74
| | | | | | |
96 61 34 15 4---5---6 21 44 75
| | | | | | | | |
95 60 33 14 3 *0* 7 22 45 76
| | | | | | | | | |
94 59 32 13 *2*--1 8 23 46 77
| | | | | | | |
93 58 31 *12*-11--10---9 24 47 78
| | | | | |
92 57 *30*-29--28--27--26--25 48 79
| | | |
91 *56*-55--54--53--52--51--50--49 80
| |
*90*-89--88--87--86--85--84--83--82--81
.
[Edited by Jon E. Schoenfield, Jan 01 2017]
MAPLE
A002939:=n->2*n*(2*n-1): seq(A002939(n), n=0..100); # Wesley Ivan Hurt, Jan 28 2017
MATHEMATICA
Table[2*n*(2*n-1), {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
2#(2#-1)&/@Range[0, 50] (* Harvey P. Dale, Mar 06 2011 *)
PROG
(PARI) a(n)=2*binomial(2*n, 2) \\ Charles R Greathouse IV, Jul 25 2011
(Magma) [2*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Jul 26 2011
(Haskell)
a002939 n = (* 2) . a000384
a002939_list = scanl1 (+) a017089_list
-- Reinhard Zumkeller, Jun 08 2015
(Python) a=lambda n: 2*n*(2*n-1) # Indranil Ghosh, Jan 01 2017
CROSSREFS
Sequences from spirals: A001107, A002939, A007742, A033951-A033953, A033954, A033989-A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A016789, A017041, A017485, A125202.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=8). - Bruno Berselli, Jun 10 2013
Cf. A017089 (first differences), A268684 (partial sums), A010050 (partial products).
Cf. A371064.
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A033996
8 times triangular numbers: a(n) = 4*n*(n+1).
+10
82
0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Write 0, 1, 2, ... in a clockwise spiral; sequence gives numbers on one of 4 diagonals.
Also, least m > n such that T(m)*T(n) is a square and more precisely that of A055112(n). {T(n) = A000217(n)}. - Lekraj Beedassy, May 14 2004
Also sequence found by reading the line from 0, in the direction 0, 8, ... and the same line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Axis perpendicular to A195146 in the same spiral. - Omar E. Pol, Sep 18 2011
Number of diagonals with length sqrt(5) in an (n+1) X (n+1) square grid. Every 1 X 2 rectangle has two such diagonals. - Wesley Ivan Hurt, Mar 25 2015
Imagine a board made of squares (like a chessboard), one of whose squares is completely surrounded by square-shaped layers made of adjacent squares. a(n) is the total number of squares in the first to n-th layer. a(1) = 8 because there are 8 neighbors to the unit square; adding them gives a 3 X 3 square. a(2) = 24 = 8 + 16 because we need 16 more squares in the next layer to get a 5 X 5 square: a(n) = (2*n+1)^2 - 1 counting the (2n+1) X (2n+1) square minus the central square. - R. J. Cano, Sep 26 2015
The three platonic solids (the simplex, hypercube, and cross-polytope) with unit side length in n dimensions all have rational volume if and only if n appears in this sequence, after 0. - Brian T Kuhns, Feb 26 2016
The number of active (ON, black) cells in the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016
The square root of a(n), n>0, has continued fraction [2n; {1,4n}] with whole number part 2n and periodic part {1,4n}. - Ron Knott, May 11 2017
Numbers k such that k+1 is a square and k is a multiple of 4. - Bruno Berselli, Sep 28 2017
a(n) is the number of vertices of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch, May 13 2018
a(n) is the number of vertices in conjoined n X n octagons which are arranged into a square array, a.k.a. truncated square tiling. - Donghwi Park, Dec 20 2020
a(n-2) is the number of ways to place 3 adjacent marks in a diagonal, horizontal, or vertical row on an n X n tic-tac-toe grid. - Matej Veselovac, May 28 2021
REFERENCES
Stuart M. Ellerstein, J. Recreational Math. 29 (3) 188, 1998.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10, 2016, 884-892.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Leo Tavares, Illustration: Centroid Diamonds.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Knight Graph.
Stephen Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.
FORMULA
a(n) = 4*n^2 + 4*n = (2*n+1)^2 - 1.
G.f.: 8*x/(1-x)^3.
a(n) = A016754(n) - 1 = 2*A046092(n) = 4*A002378(n). - Lekraj Beedassy, May 25 2004
a(n) = A049598(n) - A046092(n); a(n) = A124080(n) - A002378(n). - Zerinvary Lajos, Mar 06 2007
a(n) = 8*A000217(n). - Omar E. Pol, Dec 12 2008
a(n) = A005843(n) * A163300(n). - Juri-Stepan Gerasimov, Jul 26 2009
a(n) = a(n-1) + 8*n (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
For n > 0, a(n) = A058031(n+1) - A062938(n-1). - Charlie Marion, Apr 11 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Mar 25 2015
a(n) = A000578(n+1) - A152618(n). - Bui Quang Tuan, Apr 01 2015
a(n) - a(n-1) = A008590(n), n > 0. - Altug Alkan, Sep 26 2015
From Ilya Gutkovskiy, May 19 2016: (Start)
E.g.f.: 4*x*(2 + x)*exp(x).
Sum_{n>=1} 1/a(n) = 1/4. (End)
Product_{n>=1} a(n)/A016754(n) = Pi/4. - Daniel Suteu, Dec 25 2016
a(n) = A056220(n) + A056220(n+1). - Bruce J. Nicholson, May 29 2017
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^2. - Seiichi Manyama, Dec 23 2018
a(n)*a(n+k) + 4*k^2 = m^2 where m = (a(n) + a(n+k))/2 - 2*k^2; for k=1, m = 4*n^2 + 8*n + 2 = A060626(n). - Ezhilarasu Velayutham, May 22 2019
Sum_{n>=1} (-1)^n/a(n) = 1/4 - log(2)/2. - Vaclav Kotesovec, Dec 21 2020
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(4/Pi)*cos(Pi/sqrt(2)).
Product_{n>=1} (1 + 1/a(n)) = 4/Pi (A088538). (End)
EXAMPLE
Spiral with 0, 8, 24, 48, ... along lower right diagonal:
.
36--37--38--39--40--41--42
| |
35 16--17--18--19--20 43
| | | |
34 15 4---5---6 21 44
| | | | | |
33 14 3 0 7 22 45
| | | | \ | | |
32 13 2---1 8 23 46
| | | \ | |
31 12--11--10---9 24 47
| | \ |
30--29--28--27--26--25 48
\
[Reformatted by Jon E. Schoenfield, Dec 25 2016]
MAPLE
seq(8*binomial(n+1, 2), n=0..46); # Zerinvary Lajos, Nov 24 2006
[seq((2*n+1)^2-1, n=0..46)];
MATHEMATICA
Table[(2n - 1)^2 - 1, {n, 50}] (* Alonso del Arte, Mar 31 2013 *)
PROG
(PARI) nsqm1(n) = { forstep(x=1, n, 2, y = x*x-1; print1(y, ", ") ) }
(Magma) [ 4*n*(n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014
CROSSREFS
Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A088538, A124080, A008590 (first differences), A130809 (partial sums).
Sequences from spirals: A001107, A002939, A002943, A007742, A033951, A033952, A033953, A033954, A033988, A033989, A033990, A033991, A033996. - Omar E. Pol, Dec 12 2008
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 11 1999
STATUS
approved
A002943
a(n) = 2*n*(2*n+1).
+10
72
0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190, 8556, 8930
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and diagonal segments filled in. - Asher Auel, Jan 12 2000
In other words, the edge count of the (n+1) X (n+1) king graph. - Eric W. Weisstein, Jun 20 2017
Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals. (See Example section.)
The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as A016813(n)^2 - a(n)*2^2 = 1. - Vincenzo Librandi, Jul 20 2010 - Nov 25 2012
Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 27 2010
The hyper-Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1 <= i,j <= n, i != j} (= the complete bipartite graph K(n,n) with horizontal edges removed). The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013
Sum of the numbers from n to 3n. - Wesley Ivan Hurt, Oct 27 2014
REFERENCES
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Leo Tavares, Illustration: Twin Diamond Stars.
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 4*n^2 + 2*n.
a(n) = 2*A014105(n). - Omar E. Pol, May 21 2008
a(n) = floor((2*n + 1/2)^2). - Reinhard Zumkeller, Feb 20 2010
a(n) = A007494(n) + A173511(n) = A007742(n) + n. - Reinhard Zumkeller, Feb 20 2010
a(n) = 8*n+a(n-1) - 2 with a(0)=0. - Vincenzo Librandi, Jul 20 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 11 2011
a(n+1) = A045896(2*n+1). - Reinhard Zumkeller, Dec 12 2011
G.f.: 2*x*(3+x)/(1-x)^3. - Colin Barker, Jan 14 2012
From R. J. Mathar, Jan 15 2013: (Start)
Sum_{n>=1} 1/a(n) = 1 - log(2).
Sum_{n>=1} 1/a(n)^2 = 2*log(2) + Pi^2/6 - 3. (End)
a(n) = A118729(8*n+5). - Philippe Deléham, Mar 26 2013
a(n) = 1*A001477(n) + 2*A000217(n) + 3*A000290(n). - J. M. Bergot, Apr 23 2014
a(n) = 2 * A000217(2*n) = 2 * A014105(n). - Jon Perry, Oct 27 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 + log(2)/2 - 1. - Amiram Eldar, Feb 22 2022
a(n) = A003154(n+1) - A056220(n+1). - Leo Tavares, Mar 31 2022
E.g.f.: 2*exp(x)*x*(3 + 2*x). - Stefano Spezia, Apr 24 2024
EXAMPLE
64--65--66--67--68--69--70--71--72
|
63 36--37--38--39--40--41--42
| | |
62 35 16--17--18--19--20 43
| | | | |
61 34 15 4---5---6 21 44
| | | | | | |
60 33 14 3 0 7 22 45
| | | | | | | |
59 32 13 2---1 8 23 46
| | | | | |
58 31 12--11--10---9 24 47
| | | |
57 30--29--28--27--26--25 48
| |
56--55--54--53--52--51--50--49
MAPLE
A002943 := proc(n)
2*n*(2*n+1) ;
end proc: # R. J. Mathar, Jun 28 2013
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* Harvey P. Dale, Aug 11 2011 *)
Table[2 n (2 n + 1), {n, 0, 40}] (* Harvey P. Dale, Aug 11 2011 *)
PROG
(PARI) a(n)=2*n*(2*n+1) \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [ 4*n^2+2*n: n in [0..50]]; // Vincenzo Librandi, Nov 25 2012
(Haskell)
a002943 n = 2 * n * (2 * n + 1) -- Reinhard Zumkeller, Jan 12 2014
CROSSREFS
Cf. A001477, A007395, A007494, A007742, A014105, A016813, A033954, A045896, A046092, A054000, A118729, A173511.
Same as A033951 except start at 0.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, this sequence, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, this sequence = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003154, A056220.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Formula fixed by Reinhard Zumkeller, Apr 09 2010
STATUS
approved
A007742
a(n) = n*(4*n+1).
+10
71
0, 5, 18, 39, 68, 105, 150, 203, 264, 333, 410, 495, 588, 689, 798, 915, 1040, 1173, 1314, 1463, 1620, 1785, 1958, 2139, 2328, 2525, 2730, 2943, 3164, 3393, 3630, 3875, 4128, 4389, 4658, 4935, 5220, 5513, 5814, 6123, 6440, 6765, 7098, 7439, 7788, 8145
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Write 0,1,2,... in a clockwise spiral; sequence gives the numbers that fall on the positive y-axis. (See Example section.)
Central terms of the triangle in A126890. - Reinhard Zumkeller, Dec 30 2006
a(n)*Pi is the total length of 4 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A004770. The spiral length ratio rounded down [floor(L(n)/L(1))] is A047497. See illustration in links. - Kival Ngaokrajang, Dec 27 2013
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n; {4, 4n}]. For n=1, this collapses to [2, {4}]. - Magus K. Chu, Sep 15 2022
REFERENCES
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Emilio Apricena, A version of the Ulam spiral
Robert FERREOL, Illustration by pentagons
Kival Ngaokrajang, Illustration of 4 points circle center spiral
Leo Tavares, Illustration: Triangular Layers
G. Thimm, Emails to N. J. A. Sloane, Sep. 1994
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(5+3*x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = A033991(-n) = A074378(2*n).
a(n) = floor((n + 1/4)^2). - Reinhard Zumkeller, Feb 20 2010
a(n) = A110654(n) + A173511(n) = A002943(n) - n. - Reinhard Zumkeller, Feb 20 2010
a(n) = 8*n + a(n-1) - 3. - Vincenzo Librandi, Nov 21 2010
Sum_{n>=1} 1/a(n) = Sum_{k>=0} (-1)^k*zeta(2+k)/4^(k+1) = 0.349762131... . - R. J. Mathar, Jul 10 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=18. - Philippe Deléham, Mar 26 2013
a(n) = A118729(8n+4). - Philippe Deléham, Mar 26 2013
a(n) = A000217(3*n) - A000217(n). - Bruno Berselli, Sep 21 2016
E.g.f.: (4*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = 4 - Pi/2 - 3*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(2) + log(2) + sqrt(2)*log(1 + sqrt(2)) - 4. (End)
a(n) = A081266(n) - A000217(n). - Leo Tavares, Mar 25 2022
EXAMPLE
Part of the spiral:
.
64--65--66--67--68
|
63 36--37--38--39--40--41--42
| | |
62 35 16--17--18--19--20 43
| | | | |
61 34 15 4---5---6 21 44
| | | | | | |
60 33 14 3 0 7 22 45
| | | | | | | |
59 32 13 2---1 8 23 46
| | | | | |
58 31 12--11--10---9 24 47
| | | |
57 30--29--28--27--26--25 48
| |
56--55--54--53--52--51--50--49
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 5, 18}, 50] (* Vincenzo Librandi, Jan 29 2012 *)
Table[n(4n+1), {n, 0, 50}] (* Harvey P. Dale, Aug 10 2017 *)
PROG
(PARI) a(n)=4*n^2+n
(Magma) I:=[0, 5, 18]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012
CROSSREFS
Cf. A033991, A074378.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.
Cf. A081266.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A033991
a(n) = n*(4*n-1).
+10
69
0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335, 3570, 3813, 4064, 4323, 4590, 4865, 5148, 5439, 5738, 6045, 6360, 6683, 7014, 7353, 7700, 8055, 8418
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Write 0,1,2,... in a clockwise spiral; sequence gives numbers on negative x axis. (See illustration in Example.)
This sequence is the number of expressions x generated for a given modulus n in finite arithmetic. For example, n=1 (modulus 1) generates 3 expressions: 0+0=0(mod 1), 0-0=0(mod 1), 0*0=0(mod 1). By subtracting n from 4n^2, we eliminate the counting of those expressions that would include division by zero, which would be, of course, undefined. - David Quentin Dauthier, Nov 04 2007
From Emeric Deutsch, Sep 21 2010: (Start)
a(n) is also the Wiener index of the windmill graph D(3,n).
The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
Example: a(2)=14; indeed if the triangles are OAB and OCD, then, denoting distance by d, we have d(O,A)=d(O,B)=d(A,B)=d(O,C)=d(O,D)=d(C,D)=1 and d(A,C)=d(A,D)=d(B,C)=d(B,D)=2. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(4,n), D(5,n), and D(6,n) see A152743, A028994, and A180577, respectively. (End)
Even hexagonal numbers divided by 2. - Omar E. Pol, Aug 18 2011
For n > 0, a(n) equals the number of length 3*n binary words having exactly two 0's with the n first bits having at most one 0. For example a(2) = 14. Words are 010111, 011011, 011101, 011110, 100111, 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100. - Franck Maminirina Ramaharo, Mar 09 2018
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n-1; {1, 2, 1, 4n-2}]. For n=1, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 06 2022
REFERENCES
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Emilio Apricena, A version of the Ulam spiral.
Eric W. Weisstein, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = A007742(-n) = A074378(2n-1) = A014848(2n).
G.f.: x*(3+5*x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = A014635(n)/2. - Zerinvary Lajos, Jan 16 2007
From Zerinvary Lajos, Jun 12 2007: (Start)
a(n) = A000326(n) + A005476(n).
a(n) = A049452(n) - A001105(n). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Harvey P. Dale, Oct 10 2011
a(n) = A118729(8n+2). - Philippe Deléham, Mar 26 2013
From Ilya Gutkovskiy, Dec 04 2016: (Start)
E.g.f.: x*(3 + 4*x)*exp(x).
Sum_{n>=1} 1/a(n) = 3*log(2) - Pi/2 = 0.50864521488... (End)
a(n) = Sum_{i=n..3n-1} i. - Wesley Ivan Hurt, Dec 04 2016
From Franck Maminirina Ramaharo, Mar 09 2018: (Start)
a(n) = binomial(2*n, 2) + 2*n^2.
a(n) = A054556(n+1) - 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3-2*sqrt(2)))/sqrt(2) - log(2). - Amiram Eldar, Mar 20 2022
EXAMPLE
Clockwise spiral (with sequence terms parenthesized) begins
16--17--18--19
|
15 4---5---6
| | |
(14) (3) (0) 7
| | | |
13 2---1 8
| |
12--11--10---9
MAPLE
[seq(binomial(4*n, 2)/2, n=0..45)]; # Zerinvary Lajos, Jan 16 2007
MATHEMATICA
Table[n*(4*n - 1), {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
LinearRecurrence[{3, -3, 1}, {0, 3, 14}, 50] (* Harvey P. Dale, Oct 10 2011 *)
PROG
(PARI) a(n)=4*n^2-n;
CROSSREFS
Sequences from spirals: A001107, A002939, A007742, A033951-A033954, A033989-A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A074378, A014848, A152743, A028994, A000326, A001105, A005476, A014635, A016742, A049452, A118729.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Two remarks combined into one by Emeric Deutsch, Oct 03 2010
STATUS
approved
A033951
Write 1,2,... in a clockwise spiral; sequence gives numbers on positive x axis.
+10
57
1, 8, 23, 46, 77, 116, 163, 218, 281, 352, 431, 518, 613, 716, 827, 946, 1073, 1208, 1351, 1502, 1661, 1828, 2003, 2186, 2377, 2576, 2783, 2998, 3221, 3452, 3691, 3938, 4193, 4456, 4727, 5006, 5293, 5588, 5891, 6202, 6521, 6848, 7183, 7526, 7877, 8236, 8603, 8978
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ulam's spiral (S spoke of A054552). - Robert G. Wilson v, Oct 31 2011
a(n) is the first term in a sum of 2*n + 1 consecutive integers that equals (2*n + 1)^3. - Patrick J. McNab, Dec 24 2016
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 4*n^2 + 3*n + 1.
G.f.: (1 + 5*x + 2*x^2)/(1-x)^3.
A014848(2n+1) = a(n).
Equals A132774 * [1, 2, 3, ...]; = binomial transform of [1, 7, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 28 2007
a(n) = A016754(n) - n. - Reinhard Zumkeller, May 17 2009
a(n) = a(n-1) + 8*n-1 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
a(0)=1, a(1)=8, a(2)=23, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 07 2015
E.g.f.: exp(x)*(1 + 7*x + 4*x^2). - Stefano Spezia, Apr 24 2024
EXAMPLE
Spiral begins:
.
65--66--67--68--69--70--71--72--73
| |
64 37--38--39--40--41--42--43 74
| | | |
63 36 17--18--19--20--21 44 75
| | | | | |
62 35 16 5---6---7 22 45 76
| | | | | | | |
61 34 15 4 1 8 23 46 77
| | | | | | | |
60 33 14 3---2 9 24 47
| | | | | |
59 32 13--12--11--10 25 48
| | | |
58 31--30--29--28--27--26 49
| |
57--56--55--54--53--52--51--50
MAPLE
A033951:=n->4*n^2 + 3*n + 1: seq(A033951(n), n=0..100); # Wesley Ivan Hurt, Feb 11 2017
MATHEMATICA
lst={}; Do[p=4*n^2+3*n+1; AppendTo[lst, p], {n, 1, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 23}, 60] (* Harvey P. Dale, Feb 07 2015 *)
CoefficientList[Series[(1 + 5 x + 2 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Feb 12 2017 *)
PROG
(PARI) a(n)=4*n^2+3*n+1
(Python)
[4*n**2 + 3*n + 1 for n in range(46)] # Michael S. Branicky, Jan 08 2021
CROSSREFS
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A014848, A132774.
KEYWORD
nonn,easy,nice
AUTHOR
Olivier Gorin (gorin(AT)roazhon.inra.fr)
EXTENSIONS
Extended (with formula) by Erich Friedman
STATUS
approved
A033954
Second 10-gonal (or decagonal) numbers: n*(4*n+3).
+10
55
0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451, 3690, 3937, 4192, 4455, 4726, 5005, 5292, 5587, 5890, 6201, 6520, 6847, 7182, 7525, 7876, 8235
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Same as A033951 except start at 0. See example section.
Bisection of A074377. Also sequence found by reading the line from 0, in the direction 0, 22, ... and the line from 7, in the direction 7, 45, ..., in the square spiral whose vertices are the generalized 10-gonal numbers A074377. - Omar E. Pol, Jul 24 2012
REFERENCES
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Emilio Apricena, A version of the Ulam spiral.
Leo Tavares, Illustration: V numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = A001107(-n) = A074377(2*n).
G.f.: x*(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = a(n-1) + 8*n - 1 with a(0)=0. - Vincenzo Librandi, Jul 20 2010
For n>0, Sum_{j=0..n} (a(n) + j)^4 + (4*A000217(n))^3 = Sum_{j=n+1..2n} (a(n) + j)^4; see also A045944. - Charlie Marion, Dec 08 2007, edited by Michel Marcus, Mar 14 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 22. - Philippe Deléham, Mar 26 2013
a(n) = A118729(8n+6). - Philippe Deléham, Mar 26 2013
a(n) = A002943(n) + n = A007742(n) + 2n = A016742(n) + 3n = A033991(n) + 4n = A002939(n) + 5n = A001107(n) + 6n = A033996(n) - n. - Philippe Deléham, Mar 26 2013
Sum_{n>=1} 1/a(n) = 4/9 + Pi/6 - log(2) = 0.2748960394827980081... . - Vaclav Kotesovec, Apr 27 2016
E.g.f.: exp(x)*x*(7 + 4*x). - Stefano Spezia, Jun 08 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(2)) + log(2)/3 - 4/9 - sqrt(2)*arcsinh(1)/3. - Amiram Eldar, Nov 28 2021
For n>0, (a(n)^2 + n)/(a(n) + n) = (4*n + 1)^2/4, a ratio of two squares. - Rick L. Shepherd, Feb 23 2022
a(n) = A060544(n+1) - A000217(n+1). - Leo Tavares, Mar 31 2022
EXAMPLE
36--37--38--39--40--41--42
| |
35 16--17--18--19--20 43
| | | |
34 15 4---5---6 21 44
| | | | | |
33 14 3 0===7==22==45==76=>
| | | | | |
32 13 2---1 8 23
| | | |
31 12--11--10---9 24
| |
30--29--28--27--26--25
MATHEMATICA
Table[n(4n+3), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 7, 22}, 50] (* Harvey P. Dale, May 06 2018 *)
PROG
(PARI) a(n)=4*n^2+3*n
(Magma) [n*(4*n+3): n in [0..50]]; // G. C. Greubel, May 24 2019
(Sage) [n*(4*n+3) for n in (0..50)] # G. C. Greubel, May 24 2019
(GAP) List([0..50], n-> n*(4*n+3)) # G. C. Greubel, May 24 2019
CROSSREFS
Cf. A002620, A033951.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, this sequence, A062728, A135705.
Cf. A060544.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A033952
Write 1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 1 at the origin and 2 at x=0, y=-1; sequence gives the numbers on the positive x-axis.
+10
26
1, 8, 6, 2, 3, 6, 6, 0, 3, 1, 8, 0, 2, 7, 1, 3, 9, 3, 4, 3, 9, 6, 0, 6, 8, 9, 6, 1, 2, 4, 2, 1, 5, 9, 4, 1, 0, 7, 7, 1, 7, 8, 0, 2, 6, 3, 4, 2, 7, 1, 8, 2, 0, 2, 2, 3, 5, 6, 6, 3, 2, 4, 1, 4, 1, 5, 6, 4, 2, 9, 2, 5, 5, 6, 8, 5, 0, 6, 4, 6, 7, 9, 0, 7, 6, 6, 7, 7, 7, 6, 4, 8, 0, 9, 1, 9, 5, 5, 9, 1, 0, 4, 0, 0, 9
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Same as the South spoke of the Champernowne spiral (A244677).
LINKS
Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
EXAMPLE
The spiral begins
.
3---1---4---1---5
| |
1 5---6---7 1
| | | |
2 4 1 8 6
| | | | |
1 3---2 9 1
| | |
1---1---0---1 7
.
MATHEMATICA
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]];
f[n_] := 4n^2 - 5n + 2; Array[ almostNatural[ f@#, 10] &, 105] (* Robert G. Wilson v, Aug 08 2014 *)
CROSSREFS
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
KEYWORD
nonn,base,easy,nice
AUTHOR
Olivier Gorin (gorin(AT)roazhon.inra.fr)
EXTENSIONS
More terms from Andrew J. Gacek (andrew(AT)dgi.net)
Edited by Charles R Greathouse IV, Nov 01 2009
STATUS
approved
A033988
Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive y-axis.
+10
16
0, 5, 1, 4, 3, 7, 8, 0, 4, 7, 7, 1, 2, 6, 2, 1, 8, 7, 4, 2, 6, 1, 8, 9, 2, 7, 6, 0, 6, 5, 1, 2, 0, 4, 1, 5, 8, 5, 1, 8, 8, 8, 2, 1, 2, 3, 2, 4, 9, 0, 2, 8, 9, 9, 3, 3, 2, 0, 3, 7, 9, 3, 4, 2, 8, 8, 4, 7, 1, 5, 5, 3, 7, 4, 5, 9, 7, 5, 6, 5, 9, 8, 7, 1, 5, 3, 7, 8, 4, 0, 8, 5, 6, 9, 9, 3, 1, 0, 9, 8, 1, 1, 6, 9, 9
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
In other words, write 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... in a clockwise spiral, starting with the 0 and taking the first step south; the sequence is then picked out from the resulting spiral by starting at the origin and moving north.
LINKS
Andrew Woods, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A033307(4*n^2 + n - 1) for n > 0. - Andrew Woods, May 18 2012
EXAMPLE
1---3---1---4---1
| |
2 4---5---6 5
| | | |
1 3 0 7 1
| | | | |
1 2---1 8 6
| | |
1---0---1---9 1
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it.
Then the sequence is obtained by reading vertically upwards, starting from the initial 0.
MATHEMATICA
nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2 + n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)
CROSSREFS
Sequences based on the same spiral: A033953, A033989, A033990. Spiral without zero: A033952.
Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996.
Cf. A033307.
KEYWORD
nonn,easy,base
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Andrew Gacek (andrew(AT)dgi.net)
Edited by Jon E. Schoenfield, Aug 12 2018
STATUS
approved
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