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 A090307 #26 Sep 08 2022 08:45:12
%S A090307 2,18,326,5886,106274,1918818,34644998,625528782,11294163074,
%T A090307 203920464114,3681862517126,66477445772382,1200275886420002,
%U A090307 21671443401332418,391286257110403526,7064824071388595886
%N A090307 a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.
%C A090307 Lim_{n-> infinity} a(n)/a(n+1) = 0.0553851... = 1/(9+sqrt(82)) = (sqrt(82)-9).
%C A090307 Lim_{n-> infinity} a(n+1)/a(n) = 18.0553851... = (9+sqrt(82)) = 1/(sqrt(82)-9).
%H A090307 G. C. Greubel, <a href="/A090307/b090307.txt">Table of n, a(n) for n = 0..500</a>
%H A090307 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A090307 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A090307 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,1).
%F A090307 a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.
%F A090307 a(n) = (9+sqrt(82))^n + (9-sqrt(82))^n.
%F A090307 (a(n))^2 = a(2n) - 2 if n=1, 3, 5, ...
%F A090307 (a(n))^2 = a(2n) + 2 if n=2, 4, 6, ...
%F A090307 G.f.: (2-18*x)/(1-18*x-x^2). - _Philippe Deléham_, Nov 02 2008
%F A090307 a(n) = Lucas(n, 18) = 2*(-i)^n * ChebyshevT(n, 9*i). - _G. C. Greubel_, Dec 30 2019
%F A090307 E.g.f.: 2*exp(9*x)*cosh(sqrt(82)*x). - _Stefano Spezia_, Dec 31 2019
%e A090307 a(4) = 18*a(3) + a(2) = 18*5886 + 326 = (9+sqrt(82))^4 + (9-sqrt(82))^4 = 106273.9999905903 + 0.000009406 = 106274.
%p A090307 seq(simplify(2*(-I)^n*ChebyshevT(n, 9*I)), n = 0..20); # _G. C. Greubel_, Dec 30 2019
%t A090307 LinearRecurrence[{18,1},{2,18},25] (* or *) CoefficientList[ Series[ (2-18x)/(1-18x-x^2),{x,0,25}],x] (* _Harvey P. Dale_, Apr 22 2011 *)
%t A090307 LucasL[Range[20]-1, 18] (* _G. C. Greubel_, Dec 30 2019 *)
%o A090307 (PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 9*I) ) \\ _G. C. Greubel_, Dec 30 2019
%o A090307 (Magma) m:=18; 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 A090307 (Sage) [2*(-I)^n*chebyshev_T(n, 9*I) for n in (0..20)] # _G. C. Greubel_, Dec 30 2019
%o A090307 (GAP) m:=18;; 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 A090307 Cf. A083218, A087215.
%Y A090307 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), this sequence (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).
%K A090307 easy,nonn
%O A090307 0,1
%A A090307 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004
%E A090307 More terms from _Ray Chandler_, Feb 14 2004