A167661 Number of partitions of n into odd squares.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 25, 25, 26, 26, 28, 28, 29, 30, 31

Offset

0,10

Comments

A167662 and A167663 give record values and where they occur: A167662(n)=a(A167663(n)) and a(m) < A167662(n) for m < A167663(n).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from R. Zumkeller)

Index entries for sequences related to sums of squares.

Formula

a(n) = f(n,1,8) with f(x,y,z) = if x<y then 0^x else f(x-y,y,z)+f(x,y+z,z+8).

G.f.: G = 1/Product_{i>=1}(1-x^{(2i-1)^2}). - Emeric Deutsch , Jan 26 2016

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 4) * Zeta(3/2)^(1/3) / (4 * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017

Example

a(10)=#{9+1,1+1+1+1+1+1+1+1+1+1}=2;
a(20)=#{9+9+1+1,9+1+1+1+1+1+1+1+1+1+1+1,20x1}=3;
a(30)=#{25+1+1+1+1+1,9+9+9+1+1+1,9+9+12x1,9+21x1,30x1}=5.

Maple

g := 1/mul(1-x^((2*i-1)^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)

Crossrefs

Cf. A001156, A000009, A101412, A016754, A167700.

Sequence in context: A279951 A279224 A167383 * A187187 A300358 A102682

Adjacent sequences: A167658 A167659 A167660 * A167662 A167663 A167664

Keyword

nonn

Author

Reinhard Zumkeller, Nov 08 2009

Status

approved