This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A159673 #32 Sep 26 2022 01:51:09
%S A159673 0,56,43848,34289136,26814060560,20968561068840,16397387941772376,
%T A159673 12822736401904929248,10027363468901712899616,
%U A159673 7841385409944737582570520,6131953363213315887857247080,4795179688647403079566784646096,3749824384568905994905337736000048
%N A159673 Expansion of 56*x^2/(1 - 783*x + 783*x^2 - x^3).
%C A159673 Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j) + 1 = a(j)*a(j) and 15*n(j) + 1 = b(j)*b(j) with positive integer numbers.
%H A159673 Vincenzo Librandi, <a href="/A159673/b159673.txt">Table of n, a(n) for n = 1..200</a>
%H A159673 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (783,-783,1).
%F A159673 The a(j) recurrence is a(1)=1, a(2)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (A159668).
%F A159673 The b(j) recurrence is b(1)=1, b(2)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (A159669).
%F A159673 The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (this sequence).
%F A159673 G.f.: 56*x^2/((1-x)*(1 - 782*x + x^2)). - _Vincenzo Librandi_, Feb 26 2014
%F A159673 a(n) = -((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390. - _Colin Barker_, Jul 25 2016
%F A159673 a(n) = (14/195)*(-1 + ChebyshevU(n, 391) - 781*ChebyshevU(n-1, 391)). - _G. C. Greubel_, Sep 25 2022
%p A159673 for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
%p A159673 n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
%p A159673 # Second program
%p A159673 seq((14/195)*(simplify(ChebyshevU(n, 391) -781*ChebyshevU(n-1, 391)) -1), n=1..30); # _G. C. Greubel_, Sep 25 2022
%t A159673 CoefficientList[Series[56 x/(- x^3 + 783 x^2 - 783 x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)
%t A159673 LinearRecurrence[{783,-783,1},{0,56,43848},20] (* _Harvey P. Dale_, Jan 06 2019 *)
%o A159673 (PARI) Vec(56*x^2/(-x^3+783*x^2-783*x+1) + O(x^100)) \\ _Colin Barker_, Feb 24 2014
%o A159673 (PARI) a(n) = round(-((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390) \\ _Colin Barker_, Jul 25 2016
%o A159673 (Magma)
%o A159673 b:= func< n | Evaluate(ChebyshevSecond(n),391) >;
%o A159673 [(14/195)*(-1 +b(n+1) -781*b(n)): n in [1..30]]; // _G. C. Greubel_, Sep 25 2022
%o A159673 (SageMath)
%o A159673 def A159673(n): return (14/195)*(-1 + chebyshev_U(n, 391) - 781*chebyshev_U(n-1, 391))
%o A159673 [A159673(n) for n in range(1,30)] # _G. C. Greubel_, Sep 25 2022
%Y A159673 Cf. A157456, A159668, A159669.
%K A159673 nonn,easy
%O A159673 1,2
%A A159673 _Paul Weisenhorn_, Apr 19 2009
%E A159673 More terms and new name from _Colin Barker_, Feb 24 2014