A037214 Expansion of ( Sum_{k>=0} k*q^(k^2) )^2.

0, 0, 1, 0, 0, 4, 0, 0, 4, 0, 6, 0, 0, 12, 0, 0, 0, 8, 9, 0, 16, 0, 0, 0, 0, 24, 10, 0, 0, 20, 0, 0, 16, 0, 30, 0, 0, 12, 0, 0, 24, 40, 0, 0, 0, 36, 0, 0, 0, 0, 39, 0, 48, 28, 0, 0, 0, 0, 42, 0, 0, 60, 0, 0, 0, 72, 0, 0, 32, 0, 0, 0, 36, 48, 70, 0, 0, 0, 0, 0, 64, 0, 18, 0, 0, 120, 0, 0, 0, 80, 54, 0, 0, 0, 0, 0, 0, 72, 49

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The range of the sequence is {0, 1, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 38, 39, 40, ...}, cf. A248807. - M. F. Hasler, Oct 14 2014

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A037215, A037216, A037217.

Programs

Maple

seq(  add(T[1]*sqrt(T[2]), T in select(T->issqr(T[2]),[seq([x,n-x**2],x=1..n)]) )  , n=1..50); # Cristóbal Camarero, Oct 03 2014

PARI

N=66; q='q+O('q^N); concat([0,0], Vec( sum(n=0,N, n*q^(n^2))^2 )) \\ Joerg Arndt, Oct 13 2014

PARI

A037214(n)={my(y);sum(x=1,sqrtint(n\2),if(issquare(n-x^2,&y),x*y))*2-if(n%2==0 && issquare(n\2),n\2)} \\ M. F. Hasler, Oct 14 2014

Formula

a(n) = sum x*y for integers x,y such that x^2+y^2=n and x>0,y>=0. - Cristóbal Camarero, Oct 03 2014

If a(n)>0, then a(n)>=2*sqrt(n-1) except for a(2)=1 and a(8)=4. Proof: The extremal values a nonzero term x*y in the above sum can have is x=1, y=sqrt(n-1) in which case it occurs a second time with x,y swapped (except for x=y=1), and x=y=sqrt(n/2) in which case it may occur only once, but x*y=n/2 is larger than 2*sqrt(n-1) for n>=15. - M. F. Hasler, Oct 14 2014