A037214 - OEIS

%I #27 Sep 14 2021 09:46:07

%S 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,

%T 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,

%U 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

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

%C 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

%H Seiichi Manyama, <a href="/A037214/b037214.txt">Table of n, a(n) for n = 0..10000</a>

%F 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

%F 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

%p 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

%o (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

%o (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

%Y Cf. A037215, A037216, A037217.

%K nonn

%O 0,6

%A _N. J. A. Sloane_