Displaying 1-10 of 34 results found.
1, 1, 9, 17, 33, 49, 73, 97, 129, 161, 201, 241, 289, 337, 393, 449, 513, 577, 649, 721, 801, 881, 969, 1057, 1153, 1249, 1353, 1457, 1569, 1681, 1801, 1921, 2049, 2177, 2313, 2449, 2593, 2737, 2889, 3041, 3201, 3361, 3529, 3697, 3873, 4049, 4233, 4417, 4609
COMMENTS
(n^2)*a(n) = A275496(n) which is a triangular number. a(2n+1)/a(2n) is convergent to 1.
FORMULA
a(0) = 1; a(n) = A275496(n)/(n^2) for n > 0. a(n) = (2*n^2 + (-1)^n).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 -x +7*x^2 +x^3) / ((1 - x)^3*(1 + x)).
(End)
Sum_{n>=0} 1/a(n) = (1 + (tan(c) + coth(c))*c)/2, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022
EXAMPLE
a(5) = A275496(5)/25 = 1225/25 = 49. a(7) = A275496(7)/49 = 4753/49 = 97. a(12) = A275496(12)/144 = 41616/144 = 289.
MATHEMATICA
CoefficientList[Series[(1 - x + 7 x^2 + x^3)/((1 - x)^3 (1 + x)), {x, 0, 48}], x] (* or as defined *)
Riffle[LinearRecurrence[{3, -3, 1}, {1, 9, 33}, #], FoldList[#1 + #2 &, 1, 16 Range@ #]] &@ 25 (* Michael De Vlieger, Aug 01 2016, after Vincenzo Librandi at A081585 and Robert G. Wilson v at A069129 *)
PROG
(PARI) Vec((1-x+7*x^2+x^3)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Aug 21 2016
Total number of possible knight moves on an (n+2) X (n+2) chessboard, if the knight is placed anywhere.
+10
41
0, 16, 48, 96, 160, 240, 336, 448, 576, 720, 880, 1056, 1248, 1456, 1680, 1920, 2176, 2448, 2736, 3040, 3360, 3696, 4048, 4416, 4800, 5200, 5616, 6048, 6496, 6960, 7440, 7936, 8448, 8976, 9520, 10080, 10656, 11248, 11856, 12480, 13120, 13776
COMMENTS
16 times the triangular numbers A000217. Centered 16-gonal numbers A069129, minus 1. Also, sequence found by reading the segment (0, 16) together with the line from 16, in the direction 16, 48, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Apr 26 2008, Nov 20 2008 For n >= 1, number of permutations of n+1 objects selected from 5 objects v, w, x, y, z with repetition allowed, containing n-1 v's. Examples: at n=1, n-1=0 (i.e., zero v's), and a(1)=16 because we have ww, wx, wy, wz, xw, xx, xy, xz, yw, yx, yy, yz, zw, zx, zy, zz; at n=2, n-1=1 (i.e., one v), and there are 3 permutations corresponding to each one in the n=1 case (e.g., the single v can be inserted in any of three places in the 2-object permutation xy, yielding vxy, xvy, and xyv), so a(2) = 3*a(1) = 3*16 = 48; at n=3, n-1=2 (i.e., two v's), and a(3) = C(4,2)*a(1) = 6*16 = 96; etc. - Zerinvary Lajos, Aug 07 2008 (this needs clarification, Joerg Arndt, Feb 23 2014) Sequence found by reading the line from 0, in the direction 0, 16, ... and the same line from 0, in the direction 0, 48, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, Oct 03 2011 For n > 0, a(n) is the area of the triangle with vertices at ((n-1)^2, n^2), ((n+1)^2, (n+2)^2), and ((n+3)^2, (n+2)^2). - J. M. Bergot, May 22 2014 For n > 0, a(n) is the number of self-intersecting points in star polygon {4*(n+1)/(2*n+1)}. - Bui Quang Tuan, Mar 28 2015 Equivalently: integers k such that k$ / (k/2)! and k$ / (k/2+1)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - Bernard Schott, Dec 02 2021
FORMULA
a(n) = 8*n*(n+1).
G.f.: 16*x/(1-x)^3.
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/8.
Product_{n>=1} (1 - 1/a(n)) = -(8/Pi)*cos(sqrt(3/2)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (8/Pi)*cos(Pi/(2*sqrt(2))). (End)
EXAMPLE
3 X 3-Board: knight can be placed in 8 positions with 2 moves from each, so a(1) = 16.
MATHEMATICA
CoefficientList[Series[16 x/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 24 2014 *) LinearRecurrence[{3, -3, 1}, {0, 16, 48}, 50] (* or *) 16*Accumulate[ Range[ 0, 50]] (* Harvey P. Dale, Aug 05 2018 *)
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com), Dec 11 1999
a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.
+10
31
0, 1, 8, 17, 32, 49, 72, 97, 128, 161, 200, 241, 288, 337, 392, 449, 512, 577, 648, 721, 800, 881, 968, 1057, 1152, 1249, 1352, 1457, 1568, 1681, 1800, 1921, 2048, 2177, 2312, 2449, 2592, 2737, 2888, 3041, 3200, 3361, 3528, 3697, 3872, 4049, 4232
COMMENTS
This sequence arises from reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the triangular numbers A000217. Cf. A139591, etc. - Omar E. Pol, May 03 2008 The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=2, thus a(k) = |(P(2,0)-(-1)^k*P(2,2*k))/2|. - Peter Luschny, Jul 12 2009 Axis perpendicular to A046092 in the square spiral whose vertices are the triangular numbers A000217. See the comment above. - Omar E. Pol, Sep 14 2011 Alternating sum of the first n odd squares in decreasing order, n >= 1. Also number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton. The rules are: on the infinite square grid, start with all cells OFF, so a(0) = 0. Turn a single cell to the ON state, so a(1) = 1. At each subsequent step, the neighbor cells of each cell of the old generation are turned ON, and the cells of the old generation are turned OFF. Here "neighbor" refers to the eight adjacent cells of each ON cell. See example. - Omar E. Pol, Feb 16 2014
FORMULA
a(2n) = 8*n^2, a(2n+1) = 8*n(n+1) + 1.
a(n) = 2*n^2 + 4*n + 1 [+1 if n is odd] with a(0)=1.
G.f.: x*(x^2+6*x+1)/(1-x)^3/(1+x). (End)
Row sums of triangle A131925; binomial transform of (1, 7, 2, 4, -8, 16, -32, ...). - Gary W. Adamson, Jul 29 2007 a(n) = 2*n^2 + ((-1)^n-1)/2. - Omar E. Pol, Sep 28 2011 Sum_{n>=1} 1/a(n) = Pi^2/48 + tan(Pi/(2*sqrt(2)))*Pi /(4*sqrt(2)). - Amiram Eldar, Jan 16 2023
EXAMPLE
Illustration of initial terms as a cellular automaton:
.
. O O O O O O O
. O O O O O O O
. O O O O O O O O O O
. O O O O O O O O O O
. O O O O O O O O O O
. O O O O O O O
. O O O O O O O
.
. 1 8 17 32
.
(End)
PROG
(Haskell)
a077221 n = a077221_list !! n
a077221_list = scanl (+) 0 a047522_list
0, 8, 32, 72, 128, 200, 288, 392, 512, 648, 800, 968, 1152, 1352, 1568, 1800, 2048, 2312, 2592, 2888, 3200, 3528, 3872, 4232, 4608, 5000, 5408, 5832, 6272, 6728, 7200, 7688, 8192, 8712, 9248, 9800, 10368, 10952, 11552, 12168, 12800
COMMENTS
Opposite numbers to the centered 16-gonal numbers ( A069129) in the square spiral whose vertices are the triangular numbers ( A000217). a(n-1) is the molecular topological index of the n-wheel graph W_n. - Eric W. Weisstein, Jul 11 2011 An n X n pandiagonal magic square has a(n) orientations. - Kausthub Gudipati, Sep 15 2011 Sum of all the parts in the partitions of 4n into exactly two parts. - Wesley Ivan Hurt, Jul 23 2014 Equivalently: integers k such that k$ / (k/2-1)! and k$ / (k/2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - Bernard Schott, Dec 02 2021
FORMULA
Sum_{n>=1} 1/a(n) = Pi^2/48 ( A245058). Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/96.
Product_{n>=1} (1 + 1/a(n)) = sqrt(8)*sinh(Pi/sqrt(8))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(8)*sin(Pi/sqrt(8))/Pi. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 8, 32}, 50] (* Harvey P. Dale, Oct 05 2023 *)
Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.
+10
26
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67
COMMENTS
Row n gives the centered figurate numbers of the n-gon.
EXAMPLE
The upper left corner of the infinite array T is
|0| 1 1 1 1 1 1 1 1 1 1 ... A000012|1| 1 2 4 7 11 16 22 29 37 46 ... A000124|2| 1 3 7 13 21 31 43 57 73 91 ... A002061|3| 1 4 10 19 31 46 64 85 109 136 ... A005448|4| 1 5 13 25 41 61 85 113 145 181 ... A001844|5| 1 6 16 31 51 76 106 141 181 226 ... A005891|6| 1 7 19 37 61 91 127 169 217 271 ... A003215|7| 1 8 22 43 71 106 148 197 253 316 ... A069099|8| 1 9 25 49 81 121 169 225 289 361 ... A016754|9| 1 10 28 55 91 136 190 253 325 406 ... A060544
MATHEMATICA
T[n_, m_] := 1 + n m (m + 1)/2;
PROG
(Iverson's J language) Let cfn be the formula above. Then the first 20 rows and columns of T are: T =: cfn / ~ i. 20 where i.
AUTHOR
Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004
18-gonal (or octadecagonal) numbers: a(n) = n*(8*n-7).
+10
25
0, 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, 1261, 1470, 1695, 1936, 2193, 2466, 2755, 3060, 3381, 3718, 4071, 4440, 4825, 5226, 5643, 6076, 6525, 6990, 7471, 7968, 8481, 9010, 9555, 10116, 10693, 11286, 11895, 12520
COMMENTS
Also, sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Apr 26 2008 This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - T. D. Noe, Apr 13 2011 Also sequence found by reading the line from 0, in the direction 0, 18, ... and the parallel line from 1, in the direction 1, 51, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, Jul 18 2012 All x values for Diophantine equation 32*x + 49 = y^2 are given by this sequence and A139278. - Bruno Berselli, Nov 11 2014
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing, 2012, page 6.
FORMULA
Sum_{n>=1} 1/a(n) = ((1+sqrt(2))*Pi + 2*sqrt(2)*arccoth(sqrt(2)) + 8*log(2))/14. - Amiram Eldar, Oct 20 2020 Product_{n>=2} (1 - 1/a(n)) = 8/9. - Amiram Eldar, Jan 22 2021
Centered 24-gonal numbers.
+10
20
1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673
COMMENTS
Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011
FORMULA
a(n) = 12*n^2 - 12*n + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011 Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011 Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
EXAMPLE
a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.
MATHEMATICA
Table[12n^2-12n+1, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 73}, 50] (* Harvey P. Dale, Jul 17 2011 *)
CROSSREFS
Cf. centered k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.
0, 9, 34, 75, 132, 205, 294, 399, 520, 657, 810, 979, 1164, 1365, 1582, 1815, 2064, 2329, 2610, 2907, 3220, 3549, 3894, 4255, 4632, 5025, 5434, 5859, 6300, 6757, 7230, 7719, 8224, 8745, 9282, 9835, 10404, 10989, 11590, 12207, 12840
COMMENTS
Sequence found by reading the line from 0, in the direction 0, 9,..., in the square spiral whose vertices are the triangular numbers A000217.
FORMULA
a(n) = 8*n^2 + n.
Sequences of the form a(n) = 8*n^2+c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271- A139278, positive or negative c. - R. J. Mathar, May 12 2008 Sum_{n>=1} 1/a(n) = 8 - (1+sqrt(2))*Pi/2 - 4*log(2) - sqrt(2) * log(1+sqrt(2)) = 0.1887230016056779928... . - Vaclav Kotesovec, Sep 21 2016 G.f.: x*(7*x + 9)/(1-x)^3.
E.g.f.: (8*x^2 + 9*x)*exp(x). (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 9, 34}, 50] (* Harvey P. Dale, Apr 21 2020 *)
CROSSREFS
Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139274, A139276, A139278, A139279, A139280, A139281, A139282.
0, 2, 20, 54, 104, 170, 252, 350, 464, 594, 740, 902, 1080, 1274, 1484, 1710, 1952, 2210, 2484, 2774, 3080, 3402, 3740, 4094, 4464, 4850, 5252, 5670, 6104, 6554, 7020, 7502, 8000, 8514, 9044, 9590, 10152, 10730, 11324, 11934, 12560
COMMENTS
Sequence found by reading the line from 0, in the direction 0, 2, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A033585 in the same spiral. Twice decagonal numbers (or twice 10-gonal numbers). - Omar E. Pol, May 15 2008 a(n) is the number of walks in a cubic lattice of n dimensions that reach the point of origin for the first time after 4 steps. - Shel Kaphan, Mar 20 2023
FORMULA
a(n) = 8*n^2 - 6*n.
Sequences of the form a(n) = 8*n^2 + c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3a(n-1) - 3a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271- A139278, positive or negative c. - R. J. Mathar, May 12 2008 G.f.: (2*x)*(7*x+1)/(1-x)^3.
E.g.f.: (8*x^2 + 2*x)*exp(x). (End)
Sum_{n>=1} 1/a(n) = Pi/12 + log(2)/2. - Amiram Eldar, Mar 28 2023
MATHEMATICA
Table[8n^2-6n, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 2, 20}, 50] (* Harvey P. Dale, Sep 26 2016 *)
CROSSREFS
Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=16). - Bruno Berselli, Jun 10 2013
0, 5, 26, 63, 116, 185, 270, 371, 488, 621, 770, 935, 1116, 1313, 1526, 1755, 2000, 2261, 2538, 2831, 3140, 3465, 3806, 4163, 4536, 4925, 5330, 5751, 6188, 6641, 7110, 7595, 8096, 8613, 9146, 9695, 10260, 10841, 11438, 12051, 12680
COMMENTS
Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139277 in the same spiral. Also, sequence of numbers of the form d* A000217(n-1) + 5*n with generating functions x*(5+(d-5)*x)/(1-x)^3; the inverse binomial transform is 0,5,d,0,0,.. (0 continued). See Crossrefs. - Bruno Berselli, Feb 11 2011 Even decagonal numbers divided by 2. - Omar E. Pol, Aug 19 2011
FORMULA
a(n) = 8*n^2 - 3*n.
Sequences of the form a(n) = 8*n^2 + c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271- A139278, positive or negative c. - R. J. Mathar, May 12 2008 G.f.: x*(5 + 11*x)/(1 - x)^3.
a(0)=0, a(1)=5, a(2)=26; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 02 2012 Sum_{n>=1} 1/a(n) = 4*log(2)/3 - (sqrt(2)-1)*Pi/6 - sqrt(2)*arccoth(sqrt(2))/3. - Amiram Eldar, Jul 03 2020
MATHEMATICA
Table[n (8 n - 3), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 26}, 40] (* Harvey P. Dale, Feb 02 2012 *)
CROSSREFS
Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282. Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734.
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