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A246645 Expansion of 1/(1 - 22*x + 81*x^2), used in A246643.

%I A246645 #35 Feb 26 2025 05:53:53
%S A246645 1,22,403,7084,123205,2136706,37027927,641541208,11114644489,
%T A246645 192557340910,3335975296411,57794311907332,1001260862952013,
%U A246645 17346399720450394,300518663950795615,5206352229561021616,90197737270328030737,1562635689352773925318,27071968446864455867299
%N A246645 Expansion of 1/(1 - 22*x + 81*x^2), used in A246643.
%C A246645 This sequence is used in the formula for the curvature in a touching circle problem considered in A247512 and A246643.
%H A246645 G. C. Greubel, <a href="/A246645/b246645.txt">Table of n, a(n) for n = 0..800</a>
%H A246645 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A246645 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,-81).
%F A246645 O.g.f.: 1/(1 - 22*x + 81*x^2).
%F A246645 a(n) = 9^n*S(n, 22/9) with Chebyshev's S-polynomials (see A049310).
%F A246645 a(n) = 22*a(n-1) - 81*a(n-2), n >= 1, a(-1) = 0 and a(0) = 1.
%F A246645 a(n) = 9^n*(ap^(n+1) - am^(n+1))/(ap - am), n >= 1, with ap := (11 + 2*sqrt(10))/9 and am = 1/ap = (11 - 2*sqrt(10))/9 (Binet - de Moivre formula). a(0) = 1 (via L'Hopital's rule).
%F A246645 a(n) = 9^(n+1)*sinh(2*(n + 1)*arccsch(3))/(2*sqrt(10)). - _Federico Provvedi_, Feb 02 2021
%t A246645 CoefficientList[Series[1/(1 - 22*x + 81*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{22,-81}, {1,22}, 50] (* _G. C. Greubel_, Dec 20 2017 *)
%o A246645 (PARI) Vec(1/(1 - 22*x + 81*x^2) + O(x^40)) \\ _Michel Marcus_, Sep 30 2014
%o A246645 (Magma) I:=[1, 22]; [n le 2 select I[n] else 22*Self(n-1) - 81*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%Y A246645 Cf. A247512, A246643, A246646.
%K A246645 nonn,easy
%O A246645 0,2
%A A246645 _Wolfdieter Lang_, Sep 30 2014