A167700 Number of partitions of n into distinct odd squares.

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,131

Comments

A167701 and A167702 give record values and where they occur: A167701(n)=a(A167702(n)) and a(m) < A167701(n) for m < A167702(n);

a(A167703(n)) = 0.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Vaclav Kotesovec, Graph - The asymptotic ratio

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,z+8) + f(x,y+z,z+8).

G.f.: Product_{k>=0} (1 + x^((2*k+1)^2)). - Ilya Gutkovskiy, Jan 11 2017

a(n) ~ exp(3 * 2^(-7/3) * Pi^(1/3) * (sqrt(2)-1)^(2/3) * Zeta(3/2)^(2/3) * n^(1/3)) * (sqrt(2)-1)^(1/3) * Zeta(3/2)^(1/3) / (2^(7/6) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017

Example

a(50) = #{49+1} = 1;
a(130) = #{121+9, 81+49} = 2.

Mathematica

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

Prog

(Haskell)
a167700 = p a016754_list where
   p _  0 = 1
   p (q:qs) m = if m < q then 0 else p qs (m - q) + p qs m
-- Reinhard Zumkeller, Mar 15 2014

Crossrefs

Cf. A033461, A016754, A167661, A000700, A111900.

Sequence in context: A281815 A353633 A205988 * A010057 A359548 A359549

Adjacent sequences: A167697 A167698 A167699 * A167701 A167702 A167703

Keyword

nonn,look

Author

Reinhard Zumkeller, Nov 09 2009

Status

approved