A274324 Number of partitions of n^3 into at most two parts.

1, 1, 5, 14, 33, 63, 109, 172, 257, 365, 501, 666, 865, 1099, 1373, 1688, 2049, 2457, 2917, 3430, 4001, 4631, 5325, 6084, 6913, 7813, 8789, 9842, 10977, 12195, 13501, 14896, 16385, 17969, 19653, 21438, 23329, 25327, 27437, 29660, 32001, 34461, 37045, 39754

Offset: 0

Author

Colin Barker, Jun 18 2016

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

A subsequence of A008619.

Cf. A099392 (n^2), A274325 (n^5).

Cf. also A050492.

Programs

Magma

[(3+(-1)^n+2*n^3)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jun 25 2016

Maple

A274324:=n->(3+(-1)^n+2*n^3)/4: seq(A274324(n), n=0..50); # Wesley Ivan Hurt, Jun 25 2016

Mathematica

Table[(3+(-1)^n+2*n^3)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 25 2016 *)

PARI

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)).
b(n) = (3+(-1)^n+2*n)/4
vector(50, n, n--; b(n^3))

Formula

Coefficient of x^(n^3) in 1/((1-x)*(1-x^2)).

a(n) = A008619(n^3).

a(n) = (3+(-1)^n+2*n^3)/4.

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.

G.f.: (1-2*x+4*x^2+3*x^3) / ((1-x)^4*(1+x)).

From Stefano Spezia, Sep 28 2022: (Start)

a(n) = A050492((n+1)/2) for n odd.

E.g.f.: ((2 + x + 3*x^2 + x^3)*cosh(x) + (1 + x + 3*x^2 + x^3)*sinh(x))/2. (End)