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

%I A216283 #19 Aug 23 2017 09:50:03
%S A216283 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,
%T A216283 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,
%U A216283 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
%N A216283 Number of nonnegative solutions to the equation x^2+5*y^2 = n.
%C A216283 Records occur at 1, 9, 81, 189, 441, 1449, 3969, 12789, 13041, 30429, ... - _Antti Karttunen_, Aug 23 2017
%H A216283 Antti Karttunen, <a href="/A216283/b216283.txt">Table of n, a(n) for n = 1..65537</a>
%F A216283 G.f. T(x) * T(x^5) where T(x) = sum(n>=0, x^(n^2) ). - _Joerg Arndt_, Sep 21 2012
%e A216283 For n = 9, there are two solutions: 9 = 3^2 + 5*(0^2) = 2^2 + 5*(1^2), thus a(9) = 2.
%e A216283 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.
%o A216283 (PARI)
%o A216283 N=666;  x='x+O('x^N);
%o A216283 T(x)=sum(n=0,ceil(sqrt(N)),x^(n*n));
%o A216283 Vec(T(x)*T(x^5))
%o A216283 /* _Joerg Arndt_, Sep 21 2012 */
%o A216283 (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
%Y A216283 Cf. A033718 (all solutions x^2+5*y^2 = n).
%Y A216283 Cf. A092573, A119395, A000161, A025426, A216282.
%Y A216283 Cf. A020669 (positions of nonzeros).
%K A216283 nonn
%O A216283 1,9
%A A216283 _V. Raman_, Sep 03 2012
%E A216283 Examples from _Antti Karttunen_, Aug 23 2017