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

Author

Vladeta Jovovic, Nov 23 2006

Keywords

• easy
• nonn

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.

Crossrefs

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

Cf. A227753, A225794, A226062.

Programs

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 Sellers, Nov 30 2006

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

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

Extensions

More terms from James Sellers, Nov 30 2006