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A322889 Chebyshev T-polynomials T_n(18).

%I A322889 #26 Aug 02 2025 14:29:21
%S A322889 1,18,647,23274,837217,30116538,1083358151,38970776898,1401864610177,
%T A322889 50428155189474,1814011722210887,65253993844402458,
%U A322889 2347329766676277601,84438617606501591178,3037442904067381004807,109263505928819214581874
%N A322889 Chebyshev T-polynomials T_n(18).
%H A322889 Colin Barker, <a href="/A322889/b322889.txt">Table of n, a(n) for n = 0..600</a>
%H A322889 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A322889 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A322889 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (36,-1).
%F A322889 a(0) = 1, a(1) = 18 and a(n) = 36*a(n-1) - a(n-2) for n > 1.
%F A322889 From _Colin Barker_, Dec 30 2018: (Start)
%F A322889 G.f.: (1 - 18*x) / (1 - 36*x + x^2).
%F A322889 a(n) = ((18+sqrt(323))^(-n) * (1+(18+sqrt(323))^(2*n))) / 2. (End)
%F A322889 E.g.f.: exp(18*x)*cosh(sqrt(323)*x). - _Stefano Spezia_, Aug 02 2025
%p A322889 seq(coeff(series((1-18*x)/(1-36*x+x^2),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Dec 31 2018
%t A322889 Array[ChebyshevT[#, 18] &, 16, 0] (* or *)
%t A322889 With[{k = 18}, CoefficientList[Series[(1 - k x)/(1 - 2 k x + x^2), {x, 0, 15}], x]] (* _Michael De Vlieger_, Jan 01 2019 *)
%o A322889 (PARI) {a(n) = polchebyshev(n, 1, 18)}
%o A322889 (PARI) Vec((1 - 18*x) / (1 - 36*x + x^2) + O(x^20)) \\ _Colin Barker_, Dec 30 2018
%o A322889 (GAP) a:=[1,18];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; Print(a); # _Muniru A Asiru_, Dec 31 2018
%o A322889 (Magma) I:=[1, 18]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Jan 02 2019
%Y A322889 Column 18 of A322836.
%K A322889 nonn,easy
%O A322889 0,2
%A A322889 _Seiichi Manyama_, Dec 29 2018