A163349 a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

1, 9, 67, 463, 3089, 20241, 131363, 848087, 5459521, 35089209, 225323107, 1446179263, 9279361169, 59531488641, 381889579523, 2449671556487, 15713255235841, 100790106559209, 646496195167747, 4146789500815663

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A081180 without initial 0. Fifth binomial transform of A143095.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-23).

Crossrefs

Cf. A081180, A143095.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+2*r)*(5+r)^n+(1-2*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009

Mathematica

LinearRecurrence[{10,-23}, {1,9}, 50] (* G. C. Greubel, Dec 19 2016 *)

PARI

Vec((1-x)/(1-10*x+23*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

a(n) = ((1+2*sqrt(2))*(5+sqrt(2))^n + (1-2*sqrt(2))*(5-sqrt(2))^n)/2.

G.f.: (1-x)/(1-10*x+23*x^2).

E.g.f.: exp(5*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