A099141 a(n) = 5^n * T(n,7/5) where T is the Chebyshev polynomial of the first kind.

1, 7, 73, 847, 10033, 119287, 1419193, 16886527, 200931553, 2390878567, 28449011113, 338514191407, 4027973401873, 47928772841047, 570303484727833, 6786029465163487, 80746825394092993, 960804818888214727

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(2*n,2*k) 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 (14,-25).

Crossrefs

Column k=6 of A333988.

Cf. A081294, A001541, A090965, A083884, A099140, A099142.

Programs

Mathematica

LinearRecurrence[{14,-25},{1,7},30] (* Harvey P. Dale, Dec 26 2014 *)

Formula

G.f.: (1-7*x)/(1-14*x+25*x^2);

e.g.f.: exp(7*x)*cosh(2*sqrt(6)*x);

a(n) = 5^n * T(n, 7/5) where T is the Chebyshev polynomial of the first kind;

a(n) = Sum_{k=0..n} 6^k * binomial(2n, 2k);

a(n) = (1+sqrt(6))^(2n)/2 + (1-sqrt(6))^(2n)/2.

a(0)=1, a(1)=7, a(n) = 14*a(n-1) - 25*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009