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

0,1

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.

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 *)

Crossrefs

Cf. A004018, A330316, A330317, A330318.

Sequence in context: A297859 A298127 A220788 * A061849 A052107 A069019

Adjacent sequences: A330312 A330313 A330314 * A330316 A330317 A330318

Keyword

nonn

Author

N. J. A. Sloane, Dec 11 2019

Status

approved