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
Keywords
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].
Programs
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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