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

0, 1, 12, 128, 1344, 14080, 147456, 1544192, 16171008, 169345024, 1773404160, 18571329536, 194481487872, 2036636581888, 21327935176704, 223349036810240, 2338941478895616, 24493713157783552, 256501494231072768

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..975

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (12,-16).

Formula

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

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

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

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

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

a(n) = 2^(2*n-1)*sinh(2*n*arccsch(2))/sqrt(5). - Federico Provvedi, Feb 02 2021

Mathematica

M= {{0, 1}, {-16, 12}}; v[0] = {0, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 0, 50}] (* Roger L. Bagula, Aug 15 2006 *)
LinearRecurrence[{12, -16}, {0, 1}, 20] (* Harvey P. Dale, Sep 27 2015 *)

Prog

(Sage) [lucas_number1(n, 12, 16) for n in range(0, 19)] # Zerinvary Lajos, Apr 27 2009
(PARI) a(n) = 4^(n-1)*polchebyshev(n-1, 2, 3/2); \\ Michel Marcus, Jun 10 2018
(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

Crossrefs

Cf. A099140.

Sequence in context: A275941 A173359 A199037 * A239407 A360878 A163414

Adjacent sequences: A099154 A099155 A099156 * A099158 A099159 A099160

Keyword

easy,nonn

Author

Paul Barry, Oct 01 2004

Extensions

Name edited by Michel Marcus, Jun 10 2018

Status

approved