A225785 -id:A225785 - cp's OEIS frontend

A225786 Numbers k such that oblong(2k) + 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

Alex Ratushnyak, May 16 2013

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

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

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

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).

#include 
#include 
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;
}

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

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

a(6) from Ralf Stephan, May 17 2013

More terms from Bruno Berselli, May 18 2013

A225839 Triangular numbers representable as triangular(m) + triangular(2m).

Original entry on oeis.org

0, 378, 17766, 39209940, 1842032556, 4065365016846, 190985619471570, 421505175637435176, 19801770996209306328, 43702499616375188919330, 2053087220237987679246270, 4531162564803507161896556028, 212868189148913267563402477956, 469799997000254729943383533193910

Offset: 1

Alex Ratushnyak, May 17 2013

Triangular numbers of the sequence A147875: a(n) = A147875(A225785(n)) - see also Ralf Stephan in Program lines. [Bruno Berselli, May 20 2013]

Index entries for linear recurrences with constant coefficients, signature (1,103682,-103682,-1,1).

Cf. A000217, A147875.
Cf. A108281 (triangular numbers representable as triangular(m) + m^2).
Cf. A225785 (numbers n such that triangular(n) + triangular(2n) is a triangular number).

CoefficientList[Series[378 x (1 + 46 x + x^2)/((1 - x) (1 - 322 x + x^2) (1 + 322 x + x^2)), {x, 0, 20}], x] (* Bruno Berselli, May 20 2013 *)
LinearRecurrence[{1,103682,-103682,-1,1},{0,378,17766,39209940,1842032556},20] (* Harvey P. Dale, Jan 16 2019 *)

for(n=1,10^9,t=n*(5*n+3)/2;x=sqrtint(2*t);if(t==x*(x+1)/2,print(t))) \\ Ralf Stephan, May 17 2013

G.f.: 378*x*(1+46*x+x^2)/((1-x)*(1-322*x+x^2)*(1+322*x+x^2)). [Bruno Berselli, May 20 2013]

More terms from Bruno Berselli, May 20 2013

cp's OEIS Frontend

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

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A225839 Triangular numbers representable as triangular(m) + triangular(2m).

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cp's OEIS Frontend

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

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A225839 Triangular numbers representable as triangular(m) + triangular(2m).

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Mathematica

PARI

Formula

Extensions

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