A380481 Number of partitions of n into distinct parts less than n and not a multiple of 3.

1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 9, 11, 14, 15, 17, 20, 22, 25, 30, 33, 37, 42, 46, 52, 60, 66, 73, 82, 90, 101, 114, 125, 138, 153, 168, 186, 207, 227, 249, 274, 300, 330, 364, 398, 435, 476, 519, 568, 622, 678, 738, 804, 874, 952, 1038, 1127, 1223, 1327, 1438, 1561, 1694, 1834, 1984, 2146, 2320, 2509, 2714, 2930, 3161

Offset: 0

Author

Harman Kaur, Jun 23 2025

Keywords

• nonn

Examples

			a(3) = 1: [2,1].
a(5) = 1: [4,1].
a(6) = 2: [5,1], [4,2].
a(7) = 2: [5,2], [4,2,1].
a(8) = 2: [7,1], [5,2,1].
a(9) = 3: [8,1], [7,2], [5,4].
a(10) = 3: [8,2], [7,2,1], [5,4,1].
a(11) = 4: [10,1], [8,2,1], [7,4], [5,4,2].
		

Crossrefs

Cf. A003105, A011655, A357457.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n, n-1):
seq(a(n), n=0..74);  # Alois P. Heinz, Jul 24 2025

Mathematica

CoefficientList[
 Series[-q/QPochhammer[q, q, 1] + q^3/QPochhammer[q^3, q, 1] +
    QPochhammer[-q, q^3]*QPochhammer[-q^2, q^3], {q, 0, 500}], q]

Formula

G.f.: x^3/(1-x^3) - x/(1-x) + Product_{n>=0} (1+x^(3n+1))*(1+x^(3n+2)).

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Jul 25 2025

a(n) = A003105(n) - A011655(n). - Alois P. Heinz, Jul 25 2025