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A330315 a(n) = r(n)*r(n+1), where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.

%I A330315 #15 May 21 2024 09:40:52
%S A330315 4,16,0,0,32,0,0,0,16,32,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,96,0,0,0,0,0,
%T A330315 0,0,0,0,0,32,0,0,0,64,0,0,0,0,0,0,0,0,48,0,0,64,0,0,0,0,0,0,0,0,0,0,
%U A330315 0,64,0,0,0,0,0,0,0,32,64,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,32,0,0,96
%N A330315 a(n) = r(n)*r(n+1), where r(n) = A004018(n) is the number of ways of writing n as a sum of two squares.
%C A330315 a(n)=0 unless n == 0, 1 or 4 (mod 8).
%D A330315 H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.
%H A330315 Robert Israel, <a href="/A330315/b330315.txt">Table of n, a(n) for n = 0..10000</a>
%H A330315 Fernando Chamizo, <a href="https://doi.org/10.2969/jmsj/05110237">Correlated sums of r(n)</a>, J. Math. Soc. Japan, 51(1):237-252, 1999.
%H A330315 Fernando Chamizo, and Roberto J. Miatello, <a href="https://arxiv.org/abs/1812.10725">Sums of squares in real quadratic fields and Hilbert modular groups</a>, arXiv preprint arXiv:1812.10725 [math.NT], 2018.
%p A330315 N:= 200: # for a(0)..a(N)
%p A330315 g1:= 1 + 2*add(x^(i^2),i=1..floor(sqrt(N+1))):
%p A330315 g2:= expand(g1^2):
%p A330315 R:= [seq(coeff(g2,x,i),i=0..N+1)]:
%p A330315 seq(R[i]*R[i+1],i=1..N+1); # _Robert Israel_, Jun 12 2020
%t A330315 a[n_] := SquaresR[2, n] SquaresR[2, n + 1]; a /@ Range[0, 100] (* _Giovanni Resta_, Jun 12 2020 *)
%Y A330315 Cf. A004018, A330316, A330317, A330318.
%K A330315 nonn
%O A330315 0,1
%A A330315 _N. J. A. Sloane_, Dec 11 2019