A099142 a(n) = 6^n * T(n, 4/3) where T is the Chebyshev polynomial of the first kind.

1, 8, 92, 1184, 15632, 207488, 2757056, 36643328, 487039232, 6473467904, 86042074112, 1143628341248, 15200538791936, 202038000386048, 2685388609667072, 35692849740775424, 474411605904392192

Offset: 0

Author

Paul Barry, Sep 30 2004

Keywords

• easy
• nonn

Comments

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2n,2k) and a(n) = (1+sqrt(r+1))^(2*n)/2 + (1-sqrt(r+1))^(2*n)/2.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (16,-36).

Crossrefs

Column k=7 of A333988.

Cf. A001541, A081294, A083884, A090965, A099140, A099141.

Programs

Mathematica

LinearRecurrence[{16,-36},{1,8},20] (* Harvey P. Dale, Mar 09 2018 *)

PARI

a(n) = 6^n*polchebyshev(n, 1, 4/3); \\ Michel Marcus, Sep 08 2019

Formula

G.f.: (1-8*x)/(1-16*x+36*x^2);

E.g.f.: exp(8*x)*cosh(2*sqrt(7)*x).

a(n) = 6^n * T(n, 8/6) where T is the Chebyshev polynomial of the first kind.

a(n) = Sum_{k=0..n} 7^k * binomial(2n, 2k).

a(n) = (1+sqrt(7))^(2*n)/2 + (1-sqrt(7))^(2*n)/2.

a(0)=1, a(1)=8, a(n) = 16*a(n-1) - 36*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009