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%I #37 Sep 24 2023 13:04:15
%S 0,12,84,3960,27144,1275204,8740380,410611824,2814375312,132215732220,
%T 906220110180,42573055163112,291800061102744,13708391546789940,
%U 93958713454973484,4414059505011197664,30254413932440359200,1421313452222058857964
%N Numbers n such that triangular(n) + triangular(2*n) is a triangular number.
%C Equivalently, numbers n such that oblong(n) + oblong(2*n) is an oblong number, where oblong(n) = A002378(n) = n*(n+1).
%C Also, x values in the equation A147875(x) = A000217(y) - see _Ralf Stephan_ in Program lines. - _Bruno Berselli_, May 18 2013
%C Also, numbers m such that 2*m+1 and 10*m+1 are both squares. - _Bruno Berselli_, Mar 03 2016
%H Harvey P. Dale, <a href="/A225785/b225785.txt">Table of n, a(n) for n = 1..798</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,322,-322,-1,1).
%F 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]
%F 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]
%F a(2*n) = (Fibonacci(6*n-3)^2 + Lucas(6*n-3)*Fibonacci(6*n-1))/2. - _Greg Dresden_, Sep 24 2023
%e 12*13/2 + 24*25/2 = 27*28/2, so 12 is in the sequence.
%t 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 *)
%t LinearRecurrence[{1,322,-322,-1,1},{0,12,84,3960,27144},20] (* _Harvey P. Dale_, Apr 08 2021 *)
%o (C)
%o #include <stdio.h>
%o #include <math.h>
%o int main() {
%o unsigned long long i, s, t;
%o for (i = 0; i< (1ULL<<31); i++) {
%o s = 2*i*(2*i+1) + i*(i+1);
%o t = sqrt(s);
%o if (s==t*(t+1)) printf("%llu, ", i);
%o }
%o return 0;
%o }
%o (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 */
%Y Cf. A000217, A002378, A082183.
%Y Cf. A224419 (numbers n such that triangular(n) + triangular(2*n) is a square).
%Y Cf. A011916 (numbers n such that triangular(2*n) - triangular(n) is a triangular number).
%Y Cf. A225786 (numbers n such that oblong(2*n) + oblong(n) is a square).
%Y Cf. A225839 (triangular numbers of the form triangular(x) + triangular(2*x)).
%K nonn,easy
%O 1,2
%A _Alex Ratushnyak_, May 16 2013
%E More terms from _Bruno Berselli_, May 18 2013
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