A216282 - OEIS

%I #20 Aug 23 2017 09:49:56

%S 1,1,1,1,0,1,0,1,2,0,1,1,0,0,0,1,1,2,1,0,0,1,0,1,1,0,2,0,0,0,0,1,2,1,

%T 0,2,0,1,0,0,1,0,1,1,0,0,0,1,1,1,2,0,0,2,0,0,2,0,1,0,0,0,0,1,0,2,1,1,

%U 0,0,0,2,1,0,1,1,0,0,0,0,3,1,1,0,0,1,0,1,1,0,0,0,0,0,0,1,1,1,3,1,0,2,0,0,0,0,1,2,0,0,0,0,1,2,0,0,0,1,0,0

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

%C Records occur at 1, 9, 81, 297, 891, 1683, 5049, 15147, 31977, ... - _Antti Karttunen_, Aug 23 2017

%H Antti Karttunen, <a href="/A216282/b216282.txt">Table of n, a(n) for n = 1..65537</a>

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

%e For n = 81, there are three solutions: 81 = 9^2 + 2*(0^2) = 3^2 + 2*(6^2) = 7^2 + 2*(4^2), thus a(81) = 3.

%e For n = 65536, there is one solution: 65536 = 256^2 + 2*(0^2) = 65536 + 0, thus a(65536) = 1.

%e For n = 65537, there is one solution: 65537 = 255^2 + 2*(16^2) = 65205 + 512, thus a(65537) = 1.

%t r[n_] := Reduce[x >= 0 && y >= 0 && x^2 + 2 y^2 == n, Integers];

%t a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];

%t Table[a[n], {n, 1, 120}] (* _Jean-François Alcover_, Jun 24 2017 *)

%o (Scheme) (define (A216282 n) (cond ((< n 2) 1) (else (let loop ((k (- (A000196 n) (modulo (- n (A000196 n)) 2))) (s 0)) (if (< k 0) s (let ((x (/ (- n (* k k)) 2))) (loop (- k 2) (+ s (A010052 x))))))))) ;; _Antti Karttunen_, Aug 23 2017

%Y Cf. A092573, A119395, A000161, A025426, A216283.

%Y Cf. A002479 (positions of nonzeros), A097700 (of zeros).

%K nonn

%O 1,9

%A _V. Raman_, Sep 03 2012

%E Examples from _Antti Karttunen_, Aug 23 2017