A216283 Number of nonnegative solutions to the equation x^2+5*y^2 = n.

1, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset

1,9

Comments

Records occur at 1, 9, 81, 189, 441, 1449, 3969, 12789, 13041, 30429, ... - Antti Karttunen, Aug 23 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

G.f. T(x) * T(x^5) where T(x) = sum(n>=0, x^(n^2) ). - Joerg Arndt, Sep 21 2012

Example

For n = 9, there are two solutions: 9 = 3^2 + 5*(0^2) = 2^2 + 5*(1^2), thus a(9) = 2.
For n = 81, there are three solutions: 81  = 9^2 + 5*(0^2) = 6^2 + 5*(3^2) = 1^2 + 5*(4^2), thus a(81) = 3.

Prog

(PARI)
N=666;  x='x+O('x^N);
T(x)=sum(n=0, ceil(sqrt(N)), x^(n*n));
Vec(T(x)*T(x^5))
/* Joerg Arndt, Sep 21 2012 */
(Scheme) (define (A216283 n) (cond ((< n 2) 1) (else (let loop ((k (A000196 n)) (s 0)) (if (< k 0) s (let ((x (- n (* k k)))) (loop (- k 1) (+ s (if (zero? (modulo x 5)) (A010052 (/ x 5)) 0))))))))) ;; Antti Karttunen, Aug 23 2017

Crossrefs

Cf. A033718 (all solutions x^2+5*y^2 = n).

Cf. A092573, A119395, A000161, A025426, A216282.

Cf. A020669 (positions of nonzeros).

Sequence in context: A277143 A239434 A033770 * A262900 A242830 A101668

Adjacent sequences: A216280 A216281 A216282 * A216284 A216285 A216286

Keyword

nonn

Author

V. Raman, Sep 03 2012

Extensions

Examples from Antti Karttunen, Aug 23 2017

Status

approved