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.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Dec 11 2019

Keywords

• nonn

Comments

a(n)=0 unless n == 0, 1 or 4 (mod 8).

References

• H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Fernando Chamizo, Correlated sums of r(n), J. Math. Soc. Japan, 51(1):237-252, 1999.
• Fernando Chamizo, and Roberto J. Miatello, Sums of squares in real quadratic fields and Hilbert modular groups, arXiv preprint arXiv:1812.10725 [math.NT], 2018.

Crossrefs

Cf. A004018, A330316, A330317, A330318.

Programs

Maple

N:= 200: # for a(0)..a(N)
g1:= 1 + 2*add(x^(i^2),i=1..floor(sqrt(N+1))):
g2:= expand(g1^2):
R:= [seq(coeff(g2,x,i),i=0..N+1)]:
seq(R[i]*R[i+1],i=1..N+1); # Robert Israel, Jun 12 2020

Mathematica

a[n_] := SquaresR[2, n] SquaresR[2, n + 1]; a /@ Range[0, 100] (* Giovanni Resta, Jun 12 2020 *)