A164602 a(n) = ((1+4*sqrt(2))*(1+2*sqrt(2))^n + (1-4*sqrt(2))*(1-2*sqrt(2))^n)/2.

1, 17, 41, 201, 689, 2785, 10393, 40281, 153313, 588593, 2250377, 8620905, 32994449, 126335233, 483631609, 1851609849, 7088640961, 27138550865, 103897588457, 397765032969, 1522813185137, 5829981601057, 22319655498073

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A164703. Inverse binomial transform of A164603.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..177 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (2,7).

Crossrefs

Cf. A164703, A164603.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(1+2*r)^n+(1-4*r)*(1-2*r)^n)/2: n in [0..22] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009

Mathematica

Simplify/@Table[1/2((1-4Sqrt[2])(1-2Sqrt[2])^n+(1+2Sqrt[2])^n(1+4 Sqrt[2])),{n,0,25}] (* Harvey P. Dale, Jul 26 2011 *)

PARI

x='x+O('x^50); Vec((1+15*x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Aug 11 2017

Formula

a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 17.

G.f.: (1+15*x)/(1-2*x-7*x^2).

E.g.f.: exp(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