A163062 a(n) = ((3+sqrt(5))*(1+sqrt(5))^n + (3-sqrt(5))*(1-sqrt(5))^n)/2.

3, 8, 28, 88, 288, 928, 3008, 9728, 31488, 101888, 329728, 1067008, 3452928, 11173888, 36159488, 117014528, 378667008, 1225392128, 3965452288, 12832473088, 41526755328, 134383403008, 434873827328, 1407281266688

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

• nonn

Comments

Binomial transform of A163114. Inverse binomial transform of A163063.

Links

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

Crossrefs

Cf. A000032, A163063, A163114.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(1+r)^n+(3-r)*(1-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

Magma

I:=[3,8]; [n le 2 select I[n] else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017

Mathematica

CoefficientList[Series[(3+2*x)/(1-2*x-4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,4}, {3,8}, 30] (* G. C. Greubel, Dec 22 2017 *)

PARI

x='x+O('x^30); Vec((3+2*x)/(1-2*x-4*x^2)) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 3, a(1) = 8.

G.f.: (3+2*x)/(1-2*x-4*x^2).

a(n) = 2^n * A000032(n+2). - Diego Rattaggi, Jun 17 2020

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009