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

0,2

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).

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

Mathematica

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

Crossrefs

Column k=8 of A333988.

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

Sequence in context: A241804 A156949 A155624 * A342296 A243676 A012116

Adjacent sequences: A165221 A165222 A165223 * A165225 A165226 A165227

Keyword

easy,nonn

Author

Philippe Deléham, Sep 08 2009

Status

approved