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a(n) = 4^(n-1)*U(n-1, 3/2) where U is the Chebyshev polynomial of the second kind.
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%I #39 Jul 20 2023 14:59:52

%S 0,1,12,128,1344,14080,147456,1544192,16171008,169345024,1773404160,

%T 18571329536,194481487872,2036636581888,21327935176704,

%U 223349036810240,2338941478895616,24493713157783552,256501494231072768

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

%H G. C. Greubel, <a href="/A099157/b099157.txt">Table of n, a(n) for n = 0..975</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-16).

%F G.f.: x/(1-12*x+16*x^2).

%F E.g.f.: exp(6*x) * sinh(2*sqrt(5)*x)/sqrt(5).

%F a(n) = 12*a(n-1) - 16*a(n-2).

%F a(n) = sqrt(5)/20 * ( (sqrt(5)+1)^(2*n) - (sqrt(5)-1)^(2*n) ).

%F a(n) = Sum_{k=0..n} binomial(2*n, 2*k+1) * 5^k / 2.

%F a(n) = 2^(2*n-1)*sinh(2*n*arccsch(2))/sqrt(5). - _Federico Provvedi_, Feb 02 2021

%t M= {{0,1}, {-16,12}}; v[0] = {0,1}; v[n_]:= v[n]= M.v[n-1];

%t Table[v[n][[1]], {n,0,50}] (* _Roger L. Bagula_, Aug 15 2006 *)

%t LinearRecurrence[{12,-16},{0,1},20] (* _Harvey P. Dale_, Sep 27 2015 *)

%o (Sage) [lucas_number1(n,12,16) for n in range(0, 19)] # _Zerinvary Lajos_, Apr 27 2009

%o (PARI) a(n) = 4^(n-1)*polchebyshev(n-1, 2, 3/2); \\ _Michel Marcus_, Jun 10 2018

%o (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 Cf. A099140.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 01 2004

%E Name edited by _Michel Marcus_, Jun 10 2018