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

Author

Reinhard Zumkeller, Nov 09 2009

Keywords

• nonn
• look

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.

Examples

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

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.

Crossrefs

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

Programs

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

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 *)

Formula

a(n) = f(n,1,8) with f(x,y,z) = if x

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

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