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A225785 Numbers n such that triangular(n) + triangular(2*n) is a triangular number. 2
0, 12, 84, 3960, 27144, 1275204, 8740380, 410611824, 2814375312, 132215732220, 906220110180, 42573055163112, 291800061102744, 13708391546789940, 93958713454973484, 4414059505011197664, 30254413932440359200, 1421313452222058857964 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equivalently, numbers n such that oblong(n) + oblong(2*n) is an oblong number, where oblong(n) = A002378(n) = n*(n+1).

Also, x values in the equation A147875(x) = A000217(y) - see Ralf Stephan in Program lines. - Bruno Berselli, May 18 2013

Also, numbers m such that 2*m+1 and 10*m+1 are both squares. - Bruno Berselli, Mar 03 2016

LINKS

Table of n, a(n) for n=1..18.

Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

FORMULA

G.f.: 12*x*(1+6*x+x^2)/((1-x)*(1-18*x+x^2)(1+18*x+x^2)). [Bruno Berselli, May 18 2013]

a(n) = (1/20)*((3+(-1)^n*sqrt(5))*(2-sqrt(5))^(4*floor(n/2))+(3-(-1)^n*sqrt(5))*(2+sqrt(5))^(4*floor(n/2))-6). [Bruno Berselli, May 18 2013]

EXAMPLE

12*13/2 + 24*25/2 = 27*28/2, so 12 is in the sequence.

MATHEMATICA

CoefficientList[Series[12 x (1 + 6 x + x^2)/((1 - x) (1 - 18 x + x^2) (1 + 18 x + x^2)), {x, 0, 20}], x] (* 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+1)) printf("%llu, ", i);

  }

  return 0;

}

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

CROSSREFS

Cf. A000217, A002378, A082183.

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

Cf. A011916 (numbers n such that triangular(2*n) - triangular(n) is a triangular number).

Cf. A225786 (numbers n such that oblong(2*n) + oblong(n) is a square).

Cf. A225839 (triangular numbers of the form triangular(x) + triangular(2*x)).

Sequence in context: A155645 A095267 A118017 * A098206 A104911 A283119

Adjacent sequences:  A225782 A225783 A225784 * A225786 A225787 A225788

KEYWORD

nonn,easy

AUTHOR

Alex Ratushnyak, May 16 2013

EXTENSIONS

More terms from Bruno Berselli, May 18 2013

STATUS

approved

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)