Search: a190152 -id:a190152
|
| |
| |
|
|
|
1, 2, 12, 65, 351, 1897, 10252, 55405, 299426, 1618192, 8745217, 47261895, 255418101, 1380359512, 7459895657, 40315615410, 217878227876, 1177482265857, 6363483400447, 34390259761825, 185855747875876, 1004422742303477, 5428215467030962
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} binomial(3*n-k,3*n-3*k).
a(n) = 5*a(n-1) + 2*a(n-2) + a(n-3).
G.f.: (1-3*x)/(1-5*x-2*x^2-x^3). (End)
|
|
|
MAPLE
|
seq(add(binomial(3*n-k, 3*n-3*k), k=0..n), n=0..20);
|
|
|
MATHEMATICA
|
Table[Sum[Binomial[3n - k, 3n - 3k], {k, 0, n}], {n, 0, 22}]
LinearRecurrence[{5, 2, 1}, {1, 2, 12}, 30] (* G. C. Greubel, Dec 30 2017 *)
|
|
|
PROG
|
(Maxima) makelist(sum(binomial(3*n-k, 3*n-3*k), k, 0, n), n, 0, 22);
(PARI) x='x+O('x^30); Vec((1-3*x)/(1-5*x-2*x^2-x^3)) \\ G. C. Greubel, Dec 30 2017
(Magma) I:=[1, 2, 12]; [n le 3 select I[n] else 5*Self(n-1) +2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 30 2017
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
| |
|
|
|
1, 1, 2, 11, 30, 91, 303, 936, 2936, 9300, 29209, 91917, 289547, 911218, 2868341, 9029949, 28424456, 89477119, 281667368, 886657081, 2791106585, 8786130132, 27657838272, 87064082194, 274068969337, 862741399379, 2715822822365, 8549136143060, 26911817257385
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k,3*n-6*k).
Conjecture: G.f. ( -1+x+x^3-x^4+2*x^2 ) / ( (x^3-3*x^2+4*x-1)*(x^3+3*x^2+2*x+1) ). - R. J. Mathar, Mar 15 2013
|
|
|
MAPLE
|
seq(add(binomial(3*n-4*k, 3*n-6*k), k=0..floor(n/2)), n=0..20);
|
|
|
MATHEMATICA
|
Table[Sum[Binomial[3n-4k, 3n-6k], {k, 0, n/2}], {n, 0, 28}]
|
|
|
PROG
|
(Maxima) makelist(sum(binomial(3*n-4*k, 3*n-6*k), k, 0, n/2), n, 0, 28);
(PARI) for(n=0, 30, print1(sum(k=0, floor(n/2), binomial(3*n-4*k, 3*n-6*k)), ", ")) \\ G. C. Greubel, Dec 30 2017
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A000447
|
|
a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.
(Formerly M4697 N2006)
|
|
+10
84
|
|
|
|
0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
4 times the variance of the area under an n-step random walk: e.g., with three steps, the area can be 9/2, 7/2, 3/2, 1/2, -1/2, -3/2, -7/2, or -9/2 each with probability 1/8, giving a variance of 35/4 or a(3)/4. - Henry Bottomley, Jul 14 2003
Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch, May 30 2004
Also a(n) = (1/6)*(8*n^3-2*n), n>0: structured octagonal diamond numbers (vertex structure 9). Cf. A059722 = alternate vertex; A000447 = structured diamonds; and structured tetragonal anti-diamond numbers (vertex structure 9). Cf. A096000 = alternate vertex; A100188 = structured anti-diamonds. Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. This sequence contains the tetrahedral numbers of A000292 obtained for n= 1,3,5,7,... (see A015219). - Valentin Bakoev, Mar 03 2009
Using three consecutive numbers u, v, w, (u+v+w)^3-(u^3+v^3+w^3) equals 18 times the numbers in this sequence. - J. M. Bergot, Aug 24 2011
Number of integer solutions to 1-n <= x <= y <= z <= n-1. - Michael Somos, Dec 27 2011
Also the number of cubes in the n-th Haüy square pyramid. - Eric W. Weisstein, Sep 27 2017
|
|
|
REFERENCES
|
G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
|
|
|
FORMULA
|
a(n) = binomial(2*n+1, 3) = A000292(2*n-1).
G.f.: x*(1+6*x+x^2)/(1-x)^4.
a(n) = -a(-n) for all n in Z.
a(0)=0, a(1)=1, a(2)=10, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, May 25 2012
a(n) = v(n,n-1), where v(n,k) is the central factorial numbers of the first kind with odd indices. - Mircea Merca, Jan 25 2014
For any nonnegative integers m and n, 8*(n^3)*a(m) + 2*m*a(n) = a(2*m*n). - Ivan N. Ianakiev, Mar 04 2017
Sum_{n>=1} 1/a(n) = 6*log(2) - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)
|
|
|
EXAMPLE
|
G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
a(0) = 0, which corresponds to the empty sum.
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)
|
|
|
PROG
|
(PARI) {a(n) = n * (4*n^2 - 1) / 3};
(Haskell)
a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
(PARI) concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016
(Python)
|
|
|
CROSSREFS
|
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
A000447 is related to partitions of 2^n into powers of 2, as it is shown in the formula, example and cross-references of A002577. - Valentin Bakoev, Mar 03 2009
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Chrystal and Durell references from R. K. Guy, Apr 02 2004
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A060544
|
|
Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.
|
|
+10
57
|
|
|
|
1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
a(n) is congruent to 1 (mod 9) for all n. The sequence of digital roots of the a(n) is A000012(n). The sequence of units' digits of the a(n) is period 20: repeat [1, 0, 8, 5, 1, 6, 0, 3, 5, 6, 6, 5, 3, 0, 6, 1, 5, 8, 0, 1]. - Ant King, Jun 18 2012
Divide each side of any triangle ABC with area (ABC) into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_(2n) on side a, and similarly for sides b and c. If the hexagon with area (Hex(n)) delimited by AA_n, AA_(n+1), BB_n, BB_(n+1), CC_n and CC_(n+1) cevians, we have a(n+1) = (ABC)/(Hex(n)) for n >= 1, (see link with java applet). - Ignacio Larrosa Cañestro, Jan 02 2015; edited by Wolfdieter Lang, Jan 30 2015
For the case n = 1 see the link for Marion's Theorem (actually Marion Walter's Theorem, see the Cugo et al, reference). Also, the generalization considered here has been called there (Ryan) Morgan's Theorem. - Wolfdieter Lang, Jan 30 2015
Pollock states that every number is the sum of at most 11 terms of this sequence, but note that "1, 10, 28, 35, &c." has a typo (35 should be 55). - Michel Marcus, Nov 04 2017
a(n) is also the number of (nontrivial) paths as well as the Wiener sum index of the (n-1)-alkane graph. - Eric W. Weisstein, Jul 15 2021
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/2 = A001504(n-1)/2.
a(1-n) = a(n).
a(n) = C(n-1, 0) + 9*C(n-1, 1) + 9*C(n-1, 2); binomial transform of (1, 9, 9, 0, 0, 0, ...).
G.f.: x*(1 + 7*x + x^2)/(1-x)^3. (End)
a(n-1) = Pochhammer(4,3*n)/(Pochhammer(2,n)*Pochhammer(n+1,2*n)).
a(n-1) = 1/Hypergeometric([-3*n,3*n+3,1],[3/2,2],3/4). - Peter Luschny, Jan 09 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 9.
Sum_{n>=1} 1/a(n) = 2*Pi/(3*sqrt(3)) = A248897.
(End)
Sum_{n>=1} a(n)/n! = 11*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 11/(2*e) - 1. (End)
a(n) = P(2*n,4)*P(3*n,3)/24 for n>=2, where P(s,k) = ((s - 2)*k^2 - (s - 4)*k)/2 is the k-th s-gonal number. - Lechoslaw Ratajczak, Jul 18 2021
|
|
|
MAPLE
|
H := n -> simplify(1/hypergeom([-3*n, 3*n+3, 1], [3/2, 2], 3/4)); A060544 := n -> H(n-1); seq(A060544(i), i=1..19); # Peter Luschny, Jan 09 2012
|
|
|
MATHEMATICA
|
Take[Accumulate[Range[150]], {1, -1, 3}] (* Harvey P. Dale, Mar 11 2013 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 28}, 50] (* Harvey P. Dale, Mar 11 2013 *)
|
|
|
PROG
|
(PARI) a(n)=(3*n-1)*(3*n-2)/2
(PARI) for (n=1, 100, write("b060544.txt", n, " ", (3*n - 1)*(3*n - 2)/2); ) \\ Harry J. Smith, Jul 06 2009
(GAP) List([1..50], n->(2*n-1)^2+(n-1)*n/2); # Muniru A Asiru, Mar 01 2019
(Sage) [(3*n-1)*(3*n-2)/2 for n in (1..50)] # G. C. Greubel, Mar 02 2019
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
easy,nice,nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Formulas by Paul Berry corrected for offset 1 by Wolfdieter Lang, Jan 30 2015
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A053135
|
|
Binomial coefficients C(2*n+6,6).
|
|
+10
7
|
|
|
|
1, 28, 210, 924, 3003, 8008, 18564, 38760, 74613, 134596, 230230, 376740, 593775, 906192, 1344904, 1947792, 2760681, 3838380, 5245786, 7059052, 9366819, 12271512, 15890700, 20358520, 25827165, 32468436, 40475358, 50063860, 61474519, 74974368, 90858768
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Even-indexed members of seventh column of Pascal's triangle A007318.
Number of standard tableaux of shape (2n+1,1^6). - Emeric Deutsch, May 30 2004
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (1 + 21*x + 35*x^2 + 7*x^3)/(1-x)^7.
a(n) = binomial(2*n+6, 6) = A000579(2*n+6).
E.g.f.: (90 + 2430*x + 6975*x^2 + 5655*x^3 + 1710*x^4 + 204*x^5 + 8*x^6)* exp(x)/90. - G. C. Greubel, Sep 03 2018
Sum_{n>=0} 1/a(n) = 96*log(2) - 131/2.
Sum_{n>=0} (-1)^n/a(n) = 23/2 - 6*Pi + 12*log(2). (End)
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Table[Binomial[2*n+6, 6], {n, 0, 30}] (* G. C. Greubel, Sep 03 2018 *)
|
|
|
PROG
|
(PARI) vector(30, n, n--; binomial(2*n+6, 6)) \\ G. C. Greubel, Sep 03 2018
(Magma) [Binomial(2*n+6, 6): n in [0..30]]; // G. C. Greubel, Sep 03 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.009 seconds
|
