A035941 Number of partitions of n into parts not of the form 9k, 9k+2 or 9k-2. Also number of partitions with 1 part of size 1 and differences between parts at distance 3 are greater than 1.

1, 1, 2, 3, 4, 6, 7, 10, 13, 17, 21, 28, 35, 44, 55, 69, 84, 105, 127, 156, 189, 229, 275, 333, 397, 475, 565, 673, 795, 943, 1109, 1307, 1533, 1798, 2099, 2455, 2855, 3323, 3855, 4472, 5169, 5978, 6890, 7942, 9132, 10495, 12032, 13796, 15778, 18040

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Case k=4, i=2 of Gordon Theorem.

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Programs

Maple

# See A035937 for GordonsTheorem
A035941_list := n -> GordonsTheorem([1, 0, 1, 1, 1, 1, 0, 1, 0], n):
A035941_list(40); # Peter Luschny, Jan 22 2012

Mathematica

nmax = 60; Rest[CoefficientList[Series[Product[1 / ((1 - x^(9*k-1)) * (1 - x^(9*k-3)) * (1 - x^(9*k-4)) * (1 - x^(9*k-5)) * (1 - x^(9*k-6)) * (1 - x^(9*k-8)) ), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 12 2015 *)

Sage

# See A035937 for GordonsTheorem
def A035941_list(len) :  return GordonsTheorem([1, 0, 1, 1, 1, 1, 0, 1, 0], len)
A035941_list(40) # Peter Luschny, Jan 22 2012

Formula

a(n) ~ sin(2*Pi/9) * exp(2*Pi*sqrt(n)/3) / (3*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2015