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%I #27 Jul 10 2021 20:34:53
%S 0,1,63,3906,242110,15006915,930186621,57656563588,3573776755836,
%T 221516502298245,13730449365735355,851066344173293766,
%U 52752382889378478138,3269796672797292350791,202674641330542747270905
%N Numbers k such that 6*k and 10*k are triangular numbers.
%C Subsequence of A154293. - _Michel Marcus_, Jun 25 2014
%H Vincenzo Librandi, <a href="/A180926/b180926.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (63,-63,1).
%F a(n) = (62*a(n-1) + 1 + ((48*a(n-1) + 1)*(80*a(n-1) + 1))^(1/2))/2 with a(1)=0.
%F G.f.: -x^2 / ((x-1)*(x^2-62*x+1)). - _Colin Barker_, Jun 25 2014
%F a(n) = (-8+(4+sqrt(15))*(31+8*sqrt(15))^(-n) - (-4+sqrt(15))*(31+8*sqrt(15))^n)/480. - _Colin Barker_, Mar 03 2016
%t a[1] = 0; a[n_] := a[n] = (62 a[n - 1] + 1 + Sqrt[(48 a[n - 1] + 1)*(80 a[n - 1] + 1)])/2; Array[a, 14] (* _Robert G. Wilson v_, Sep 27 2010 *)
%t Rest[CoefficientList[Series[-x^2/((x - 1) (x^2 - 62 x + 1)), {x, 0, 30}], x]] (* _Vincenzo Librandi_, Jun 26 2014 *)
%t LinearRecurrence[{63,-63,1},{0,1,63},20] (* _Harvey P. Dale_, Dec 25 2019 *)
%o (PARI) isok(n) = ispolygonal(6*n, 3) && ispolygonal(10*n, 3); \\ _Michel Marcus_, Jun 25 2014
%K easy,nonn
%O 1,3
%A _Vladimir Pletser_, Sep 25 2010
%E a(8) onwards from _Robert G. Wilson v_, Sep 27 2010
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