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 A090313 #20 Sep 08 2022 08:45:12
%S A090313 2,22,486,10714,236194,5206982,114789798,2530582538,55787605634,
%T A090313 1229857906486,27112661548326,597708411969658,13176697724880802,
%U A090313 290485058359347302,6403847981630521446,141175140654230819114
%N A090313 a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.
%C A090313 Lim_{n-> infinity} a(n)/a(n+1) = 0.045361... = 1/(11+sqrt(122)) = (sqrt(122)-11).
%C A090313 Lim_{n-> infinity} a(n+1)/a(n) = 22.045361... = (11+sqrt(122)) = 1/(sqrt(122)-11).
%H A090313 G. C. Greubel, <a href="/A090313/b090313.txt">Table of n, a(n) for n = 0..500</a>
%H A090313 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A090313 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A090313 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,1).
%F A090313 a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.
%F A090313 a(n) = (11+sqrt(122))^n + (11-sqrt(122))^n.
%F A090313 (a(n))^2 = a(2n) - 2 if n=1, 3, 5...,
%F A090313 (a(n))^2 = a(2n) + 2 if n=2, 4, 6....
%F A090313 G.f.: (2-22*x)/(1-22*x-x^2). - _Philippe Deléham_, Nov 02 2008
%F A090313 a(n) = Lucas(n, 22) = 2*(-i)^n * ChebyshevT(n, 11*i). - _G. C. Greubel_, Dec 30 2019
%e A090313 a(4) = 236194 = 22*a(3) + a(2) = 22*10714 + 486 = (11 + sqrt(122))^4 + (11 - sqrt(122))^4 = 236193.999995766 + 0.000004233 = 236194.
%p A090313 seq(simplify(2*(-I)^n*ChebyshevT(n, 11*I)), n = 0..20); # _G. C. Greubel_, Dec 30 2019
%t A090313 LucasL[Range[20]-1,22] (* _G. C. Greubel_, Dec 29 2019 *)
%o A090313 (PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 11*I) ) \\ _G. C. Greubel_, Dec 30 2019
%o A090313 (Magma) m:=22; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 30 2019
%o A090313 (Sage) [2*(-I)^n*chebyshev_T(n, 11*I) for n in (0..20)] # _G. C. Greubel_, Dec 30 2019
%o A090313 (GAP) m:=22;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # _G. C. Greubel_, Dec 30 2019
%Y A090313 Cf. A079219.
%Y A090313 Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), this sequence (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).
%K A090313 easy,nonn
%O A090313 0,1
%A A090313 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004
%E A090313 More terms from _Ray Chandler_, Feb 14 2004