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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
Programs
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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
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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
Comments