A163350 a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

1, 6, 34, 188, 1028, 5592, 30344, 164464, 890896, 4824672, 26124832, 141453248, 765878336, 4146681216, 22451153024, 121555687168, 658129355008, 3563255219712, 19292230787584, 104452273224704, 565526954771456

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A102285. Fourth binomial transform of A163403. Inverse binomial transform of A163346.

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. A102285, A163403, A163346.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009

Mathematica

LinearRecurrence[{8,-14},{1,6},30] (* Harvey P. Dale, May 08 2014 *)

PARI

Vec((1-2*x)/(1-8*x+14*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

a(n) = ((1+sqrt(2))*(4+sqrt(2))^n+(1-sqrt(2))*(4-sqrt(2))^n)/2.

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

E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009

New name from G. C. Greubel, Dec 19 2016