A163610 a(n) = ((5 + 2*sqrt(2))*(4 + sqrt(2))^n + (5 - 2*sqrt(2))*(4 - sqrt(2))^n)/2.

5, 24, 122, 640, 3412, 18336, 98920, 534656, 2892368, 15653760, 84736928, 458742784, 2483625280, 13446603264, 72802072192, 394164131840, 2134084044032, 11554374506496, 62557819435520, 338701312393216, 1833801027048448

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

• nonn

Comments

Binomial transform of A163609. Fourth binomial transform of A163888. Inverse binomial transform of A163611.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-14).

Crossrefs

Cf. A163609, A163611, A163888.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(4+r)^n+(5-2*r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

Mathematica

LinearRecurrence[{8,-14},{5,24},30] (* Harvey P. Dale, May 29 2016 *)

PARI

x='x+O('x^50); Vec((5-16*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 24.

G.f.: (5-16*x)/(1-8*x+14*x^2).

E.g.f.: exp(4*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009