A164589 a(n) = ((4 + 3*sqrt(2))*(1 + 2*sqrt(2))^n + (4 - 3*sqrt(2))*(1 - 2*sqrt(2))^n)/8.

1, 4, 15, 58, 221, 848, 3243, 12422, 47545, 182044, 696903, 2668114, 10214549, 39105896, 149713635, 573168542, 2194332529, 8400844852, 32162017407, 123129948778, 471394019405, 1804697680256, 6909153496347, 26451190754486, 101266455983401, 387691247248204

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A096886. Inverse binomial transform of A086347.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (2,7).

Crossrefs

Cf. A086347, A096886.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(1+2*r)^n+(4-3*r)*(1-2*r)^n)/8: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

Mathematica

CoefficientList[Series[(1+2x)/(1-2x-7x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,7},{1,4},30] (* Harvey P. Dale, Jun 22 2011 *)

PARI

Vec((1+2*x)/(1-2*x-7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

Formula

a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 4.

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

E.g.f.: (1/4)*exp(x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus and R. J. Mathar, Aug 24 2009