A123975 Number of Garden of Eden partitions of n in Bulgarian Solitaire.

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 147, 190, 243, 311, 394, 499, 627, 786, 980, 1220, 1510, 1865, 2294, 2816, 3443, 4202, 5110, 6203, 7507, 9067, 10923, 13135, 15755, 18865, 22540, 26885, 32001, 38032, 45112, 53430, 63171

Offset

1,5

Comments

a(n) gives the number of times n occurs in A225794. - Antti Karttunen, Jul 27 2013

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Brian Hopkins, Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80.

Brian Hopkins and James A. Sellers, Exact enumeration of Garden of Eden partitions, INTEGERS: Electronic Journal of Combinatorial Number Theory 7(2) (2007), #A19.

Formula

a(n) = A064173(n) - A101198(n).

a(n) = Sum_{j>=1} (-1)^(j+1)*p(n-b(j)) where b(j) = 3*j*(j+1)/2 (A045943) and p(n) is the number of partitions of n (see A000041). See Hopkins & Sellers. - Michel Marcus, Sep 26 2018

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 19*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

Maple

p:=product(1/(1-q^i), i=1..200)*sum((-1)^(r-1)*q^((3*r^2+3*r)/2), r=1..200):s:=series(p, q, 200): for j from 0 to 199 do printf(`%d, `, coeff(s, q, j)) od: # James A. Sellers, Nov 30 2006

Prog

(PARI) my(N=50, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k+1)/2)))) \\ Seiichi Manyama, May 21 2023

Crossrefs

Cf. A000041, A045943, A064173, A101198, A237831.

Cf. A227753, A225794, A226062.

Sequence in context: A104503 A027340 A000701 * A321728 A214077 A094984

Adjacent sequences: A123972 A123973 A123974 * A123976 A123977 A123978

Keyword

easy,nonn

Author

Vladeta Jovovic, Nov 23 2006

Extensions

More terms from James A. Sellers, Nov 30 2006

Status

approved