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

Author

Paul Barry, Oct 01 2004

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A099140.

Programs

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

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 *)

PARI

a(n) = 4^(n-1)*polchebyshev(n-1, 2, 3/2); \\ Michel Marcus, Jun 10 2018

Sage

[lucas_number1(n,12,16) for n in range(0, 19)] # Zerinvary Lajos, Apr 27 2009

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

Extensions

Name edited by Michel Marcus, Jun 10 2018