A225786 Numbers k such that oblong(2*k) + oblong(k) is a square, where oblong(k) = A002378(k) = k*(k+1).

0, 48, 15552, 5007792, 1612493568, 519217921200, 167186558132928, 53833552500881712, 17334236718725778432, 5581570389877199773488, 1797248331303739601284800, 578708381109414274413932208, 186342301468900092621684886272

Offset

1,2

Comments

Numbers k such that k*(5*k+3) is a perfect square. Apparently a(n) = 323*a(n-1) -323*a(n-2) +a(n-3). - R. J. Mathar, May 18 2013

Links

Table of n, a(n) for n=1..13.

Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Formula

G.f.: 48*x*(1+x)/((1-x)*(1-322*x+x^2)). - Bruno Berselli, May 18 2013

a(n) = (3/20)*((2-sqrt(5))^(4n-4)+(2+sqrt(5))^(4n-4)-2). - Bruno Berselli, May 18 2013

Example

48*49 + 96*97 = 108^2, so 48 is in the sequence.

Mathematica

LinearRecurrence[{323, -323, 1}, {0, 48, 15552}, 15] (* Bruno Berselli, May 18 2013 *)

Prog

(C)
#include <stdio.h>
#include <math.h>
int main() {
  unsigned long long i, s, t;
  for (i = 0; i< (1ULL<<31); i++) {
    s = 2*i*(2*i+1) + i*(i+1);
    t = sqrt(s);
    if (s==t*t) printf("%llu, ", i);
  }
  return 0;
}

Crossrefs

Cf. A002378.

Cf. A098301 (numbers n such that oblong(2*n) - oblong(n) is a square).

Cf. A224419 (triangular(2*n) + triangular(n) is a square).

Cf. A220186 (triangular(2*n) - triangular(n) is a square).

Cf. A225785 (oblong(2*n) + oblong(n) is an oblong number).

Sequence in context: A123478 A275570 A307618 * A275454 A202928 A265866

Adjacent sequences: A225783 A225784 A225785 * A225787 A225788 A225789

Keyword

nonn,easy

Author

Alex Ratushnyak, May 16 2013

Extensions

a(6) from Ralf Stephan, May 17 2013

More terms from Bruno Berselli, May 18 2013

Status

approved