A163614 a(n) = ((1 + 3*sqrt(2))*(3 + sqrt(2))^n + (1 - 3*sqrt(2))*(3 - sqrt(2))^n)/2.

1, 9, 47, 219, 985, 4377, 19367, 85563, 377809, 1667913, 7362815, 32501499, 143469289, 633305241, 2795546423, 12340141851, 54472026145, 240451163913, 1061402800463, 4685258655387, 20681732329081, 91293583386777

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A163613. Third binomial transform of A163864. Inverse binomial transform of A163615.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A163613, A163864, A163615.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(3+r)^n+(1-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

Mathematica

LinearRecurrence[{6,-7},{1,9},30] (* Harvey P. Dale, Sep 24 2015 *)

PARI

x='x+O('x^50); Vec((1+3*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

G.f.: (1+3*x)/(1-6*x+7*x^2).

E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

Extensions

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