A163608 a(n) = ((5 + 2*sqrt(2))*(2 + sqrt(2))^n + (5 - 2*sqrt(2))*(2 - sqrt(2))^n)/2.

5, 14, 46, 156, 532, 1816, 6200, 21168, 72272, 246752, 842464, 2876352, 9820480, 33529216, 114475904, 390845184, 1334428928, 4556025344, 15555243520, 53108923392, 181325206528, 619082979328, 2113681504256, 7216560058368

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A163607. Inverse binomial transform of A163609.

Links

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

Crossrefs

Cf. A163607, A163609.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(2+r)^n+(5-2*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},{5,14},30] (* Harvey P. Dale, Jan 31 2017 *)

PARI

x='x+O('x^50); Vec((5-6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 5, a(1) = 14.

G.f.: (5-6*x)/(1-4*x+2*x^2).

a(n) = 5*A007070(n) - 6*A007070(n-1). - R. J. Mathar, Nov 08 2013

E.g.f.: exp(2*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Extensions

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