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 A154245 #45 Sep 08 2022 08:45:40
%S A154245 1,8,55,368,2449,16280,108199,719072,4778785,31758632,211059991,
%T A154245 1402652240,9321678001,61949553848,411701328775,2736064645568,
%U A154245 18183205205569,120841059834440,803079631825399,5337067516093232
%N A154245 a(n) = ( (4 + sqrt(7))^n - (4 - sqrt(7))^n )/(2*sqrt(7)).
%C A154245 Second binomial transform of A109115.
%C A154245 Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(7) = 6.6457513110....
%H A154245 Vincenzo Librandi, <a href="/A154245/b154245.txt">Table of n, a(n) for n = 1..100</a>
%H A154245 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-9).
%F A154245 From _Philippe Deléham_, Jan 06 2009: (Start)
%F A154245 a(n) = 8*a(n-1) - 9*a(n-2) for n > 1, with a(0)=0, a(1)=1.
%F A154245 G.f.: x/(1 - 8*x + 9*x^2). (End)
%F A154245 a(n) = b such that (3^(n-1)/2)*Integral_{x=0..Pi/2} (sin(n*x))/(4/3-cos(x)) dx = c + b*log(2). - _Francesco Daddi_, Aug 02 2011
%F A154245 E.g.f.: (1/sqrt(7))*exp(4*x)*sinh(sqrt(7)*x). - _G. C. Greubel_, Sep 07 2016
%t A154245 Table[Simplify[((4+Sqrt[7])^n -(4-Sqrt[7])^n)/(2*Sqrt[7])], {n, 30}] (* or *) LinearRecurrence[{8, -9},{1, 8}, 30] (* _G. C. Greubel_, Sep 07 2016 *)
%t A154245 Rest@ CoefficientList[Series[x/(1 -8x +9x^2), {x, 0, 30}], x] (* _Michael De Vlieger_, Sep 08 2016 *)
%o A154245 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 07 2009
%o A154245 (Magma) I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-9*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 08 2016
%o A154245 (Sage) [lucas_number1(n,8,9) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o A154245 (PARI) my(x='x+O('x^30)); Vec( x/(1-8*x+9*x^2) ) \\ _G. C. Greubel_, May 21 2019
%o A154245 (GAP) a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # _G. C. Greubel_, May 21 2019
%Y A154245 Equals (A094432 without initial term 0)/3.
%Y A154245 Cf. A010465 (decimal expansion of square root of 7), A109115.
%K A154245 nonn,easy
%O A154245 1,2
%A A154245 Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%E A154245 Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009
%E A154245 Edited by _Klaus Brockhaus_, Oct 06 2009