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 A225785 Numbers n such that triangular(n) + triangular(2*n) is a triangular number. 3
 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 Harvey P. Dale, Table of n, a(n) for n = 1..798 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 *) LinearRecurrence[{1, 322, -322, -1, 1}, {0, 12, 84, 3960, 27144}, 20] (* Harvey P. Dale, Apr 08 2021 *) PROG (C) #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+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: A336907 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 September 27 10:28 EDT 2021. Contains 347689 sequences. (Running on oeis4.)