A163613 a(n) = ((1 + 3*sqrt(2))*(2 + sqrt(2))^n + (1 - 3*sqrt(2))*(2 - sqrt(2))^n)/2.

1, 8, 30, 104, 356, 1216, 4152, 14176, 48400, 165248, 564192, 1926272, 6576704, 22454272, 76663680, 261746176, 893657344, 3051137024, 10417233408, 35566659584, 121432171520, 414595366912, 1415517124608, 4832877764608

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A048694. Second binomial transform of A163864. Inverse binomial transform of A163614.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2).

Crossrefs

Cf. A048694, A163864 (1, 6, 2, 12, 4, 24, ...), A163614.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+r)^n+(1-3*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

Mathematica

LinearRecurrence[{4, -2}, {1, 8}, 50] (* G. C. Greubel, Jul 30 2017 *)

PARI

x='x+O('x^50); Vec((1+4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

G.f.: (1+4*x)/(1-4*x+2*x^2).

E.g.f.: exp(2*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