A164603 a(n) = ((1+4*sqrt(2))*(2+2*sqrt(2))^n + (1-4*sqrt(2))*(2-2*sqrt(2))^n)/2.

1, 18, 76, 376, 1808, 8736, 42176, 203648, 983296, 4747776, 22924288, 110688256, 534450176, 2580553728, 12460015616, 60162277376, 290489171968, 1402605797376, 6772379877376, 32699942699008, 157889290305536

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A164602. Second binomial transform of A164702. Inverse binomial transform of A164604.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..164 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (4,4).

Crossrefs

Cf. A164602, A164702, A164604.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+2*r)^n+(1-4*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009

Mathematica

CoefficientList[Series[(-1-14 n)/(-1+4 n+4 n^2),{n,0,20}],n]  (* Harvey P. Dale, Feb 22 2011 *)

PARI

Vec((1+14*x)/(1-4*x-4*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2011

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 18.

G.f.: (1+14*x)/(1-4*x-4*x^2).

E.g.f.: exp(2*x)*( cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x) ). - G. C. Greubel, Aug 11 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009