A165224 a(0)=1, a(1)=9, a(n) = 18*a(n-1) - 49*a(n-2) for n > 1.

1, 9, 113, 1593, 23137, 338409, 4957649, 72655641, 1064876737, 15607654857, 228758827313, 3352883803641, 49142725927201, 720277760311209, 10557006115168913, 154732499817791193, 2267891697076964737

Offset: 0

Author

Philippe Deléham, Sep 08 2009

Keywords

• easy
• nonn

Comments

a(n)/a(n-1) tends to 9 + 4*sqrt(2) = 14.65685424... - Klaus Brockhaus, Sep 25 2009

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (18,-49).

Crossrefs

Column k=8 of A333988.

Cf. A081294, A001541, A090965, A083884, A099140, A099141, A099142, A026244.

Programs

Mathematica

LinearRecurrence[{18,-49},{1,9},20] (* Harvey P. Dale, Sep 30 2016 *)

Formula

G.f.: (1-9x)/(1-18x+49x^2);

e.g.f.: exp(9x)*cosh(4*sqrt(2)x);

a(n) = Sum_{k=0..n} 8^k*binomial(2n,2k) = Sum_{k=0..n} 8^k*A086645(n,k);

a(n) = 7^n*T(n,9/7) where T is the Chebyshev polynomial of the first kind;

a(n) = (1+sqrt(8))^(2n)/2 + (1-sqrt(8))^(2n)/2.

a(n) = ((9-4*sqrt(2))^n + (9+4*sqrt(2))^n)/2. - Klaus Brockhaus, Sep 25 2009