A216283 -id:A216283 - cp's OEIS frontend

A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

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, 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, 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

Offset: 1

V. Raman, Sep 03 2012

nonn

Records occur at 1, 9, 81, 297, 891, 1683, 5049, 15147, 31977, ... - Antti Karttunen, Aug 23 2017

			For n = 9, there are two solutions: 9 = 9^2 + 2*(0^2) = 1^2 + 2*(2^2), thus a(9) = 2.
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.
For n = 65536, there is one solution: 65536 = 256^2 + 2*(0^2) = 65536 + 0, thus a(65536) = 1.
For n = 65537, there is one solution: 65537 = 255^2 + 2*(16^2) = 65205 + 512, thus a(65537) = 1.

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

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

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

r[n_] := Reduce[x >= 0 && y >= 0 && x^2 + 2 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2017 *)

(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

Examples from Antti Karttunen, Aug 23 2017

A374288 a(n) is the smallest nonnegative integer k where there are exactly n nonnegative integer solutions to x^2 + 5*y^2 = k.

Original entry on oeis.org

2, 0, 9, 81, 189, 441, 1449, 10206, 3969, 12789, 13041, 35721, 30429, 194481, 117369, 115101, 186921, 2893401, 273861

Offset: 0

Seiichi Manyama, Jul 02 2024

For n > 1, a(n) is the smallest nonnegative k such that A216283(k) = n.
a(20) = 378189.
a(21) = 1035909.
a(22) = 9506889.
a(23) = 2099601.
a(24) = 882441.
a(25) = 5639949.
a(26) = 85562001.
a(27) = 6765381.
a(28) = 3403701.
a(30) = 3925341.
a(32) = 5420709.
a(33) = 83908629.
a(35) = 50759541.
a(36) = 7941969.
a(38) = 102880449.
a(40) = 15505749.
a(41) = 170067681.
a(42) = 35328069.

Cf. A374285, A374286.

Cf. A216283.

b(n, k) = sum(i=0, sqrtint(n), sum(j=0, sqrtint(n\k), i^2+k*j^2==n));
a(n, k=5) = my(cnt=0); while(b(cnt, k)!=n, cnt++); cnt;

cp's OEIS Frontend

A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

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