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

%I #19 Aug 23 2017 09:50:03

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

%C Records occur at 1, 9, 81, 189, 441, 1449, 3969, 12789, 13041, 30429, ... - _Antti Karttunen_, Aug 23 2017

%H Antti Karttunen, <a href="/A216283/b216283.txt">Table of n, a(n) for n = 1..65537</a>

%F G.f. T(x) * T(x^5) where T(x) = sum(n>=0, x^(n^2) ). - _Joerg Arndt_, Sep 21 2012

%e For n = 9, there are two solutions: 9 = 3^2 + 5*(0^2) = 2^2 + 5*(1^2), thus a(9) = 2.

%e 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 (PARI)

%o N=666; x='x+O('x^N);

%o T(x)=sum(n=0,ceil(sqrt(N)),x^(n*n));

%o Vec(T(x)*T(x^5))

%o /* _Joerg Arndt_, Sep 21 2012 */

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

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

%Y Cf. A020669 (positions of nonzeros).

%K nonn

%O 1,9

%A _V. Raman_, Sep 03 2012

%E Examples from _Antti Karttunen_, Aug 23 2017

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)