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 A159669 #37 Sep 26 2022 01:50:57
%S A159669 1,29,811,22679,634201,17734949,495944371,13868707439,387827863921,
%T A159669 10845311482349,303280893641851,8481019710489479,237165271000063561,
%U A159669 6632146568291290229,185462938641156062851,5186330135384078469599,145031780852113041085921
%N A159669 Expansion of x*(1 + x)/(1 - 28*x + x^2).
%C A159669 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 A159669 Vincenzo Librandi, <a href="/A159669/b159669.txt">Table of n, a(n) for n = 1..200</a>
%H A159669 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (28,-1).
%F A159669 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 A159669 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, ... (this sequence).
%F A159669 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, ... (A159673).
%F A159669 G.f.: x*(1+x)/(1-28*x+x^2). - _Vincenzo Librandi_, Feb 26 2014
%F A159669 a(n) = (14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26. - _Colin Barker_, Jul 25 2016
%F A159669 a(n) = chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14) = A097311(n) + A097311(n-1). - _G. C. Greubel_, Sep 25 2022
%p A159669 for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
%p A159669 n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
%p A159669 # Second program
%p A159669 seq(simplify(ChebyshevU(n, 14) +ChebyshevU(n-1, 14)), n=1..30); # _G. C. Greubel_, Sep 25 2022
%t A159669 CoefficientList[Series[(1+x)/(1-28x+x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)
%t A159669 LinearRecurrence[{28,-1},{1,29},20] (* _Harvey P. Dale_, Jul 01 2019 *)
%o A159669 (PARI) Vec(x*(x+1)/(x^2-28*x+1) + O(x^100)) \\ _Colin Barker_, Feb 24 2014
%o A159669 (PARI) a(n) = round((14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26) \\ _Colin Barker_, Jul 25 2016
%o A159669 (Magma)
%o A159669 A097311:= func< n | Evaluate(ChebyshevSecond(n-1), 14) >;
%o A159669 [A097311(n+1) + A097311(n): n in [1..30]]; // _G. C. Greubel_, Sep 25 2022
%o A159669 (SageMath)
%o A159669 def A159669(n): return chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14)
%o A159669 [A159669(n) for n in range(1,30)] # _G. C. Greubel_, Sep 25 2022
%Y A159669 Cf. A097311, A157456, A159668, A159673.
%K A159669 nonn,easy
%O A159669 1,2
%A A159669 _Paul Weisenhorn_, Apr 19 2009
%E A159669 More terms and new name from _Colin Barker_, Feb 24 2014