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A099157 a(n) = 4^(n-1)*U(n-1, 3/2) where U is the Chebyshev polynomial of the second kind.

%I A099157 #39 Jul 20 2023 14:59:52
%S A099157 0,1,12,128,1344,14080,147456,1544192,16171008,169345024,1773404160,
%T A099157 18571329536,194481487872,2036636581888,21327935176704,
%U A099157 223349036810240,2338941478895616,24493713157783552,256501494231072768
%N A099157 a(n) = 4^(n-1)*U(n-1, 3/2) where U is the Chebyshev polynomial of the second kind.
%H A099157 G. C. Greubel, <a href="/A099157/b099157.txt">Table of n, a(n) for n = 0..975</a>
%H A099157 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A099157 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-16).
%F A099157 G.f.: x/(1-12*x+16*x^2).
%F A099157 E.g.f.: exp(6*x) * sinh(2*sqrt(5)*x)/sqrt(5).
%F A099157 a(n) = 12*a(n-1) - 16*a(n-2).
%F A099157 a(n) = sqrt(5)/20 * ( (sqrt(5)+1)^(2*n) - (sqrt(5)-1)^(2*n) ).
%F A099157 a(n) = Sum_{k=0..n} binomial(2*n, 2*k+1) * 5^k / 2.
%F A099157 a(n) = 2^(2*n-1)*sinh(2*n*arccsch(2))/sqrt(5). - _Federico Provvedi_, Feb 02 2021
%t A099157 M= {{0,1}, {-16,12}}; v[0] = {0,1}; v[n_]:= v[n]= M.v[n-1];
%t A099157 Table[v[n][[1]], {n,0,50}] (* _Roger L. Bagula_, Aug 15 2006 *)
%t A099157 LinearRecurrence[{12,-16},{0,1},20] (* _Harvey P. Dale_, Sep 27 2015 *)
%o A099157 (Sage) [lucas_number1(n,12,16) for n in range(0, 19)] # _Zerinvary Lajos_, Apr 27 2009
%o A099157 (PARI) a(n) = 4^(n-1)*polchebyshev(n-1, 2, 3/2); \\ _Michel Marcus_, Jun 10 2018
%o A099157 (Magma) [n le 2 select n-1 else 12*Self(n-1) -16*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jul 20 2023
%Y A099157 Cf. A099140.
%K A099157 easy,nonn
%O A099157 0,3
%A A099157 _Paul Barry_, Oct 01 2004
%E A099157 Name edited by _Michel Marcus_, Jun 10 2018