A163062 - OEIS

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

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OFFSET

0,1

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).

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

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 *)

PROG

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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

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

CROSSREFS

Cf. A000032, A163063, A163114.

Sequence in context: A332727 A148845 A148846 * A148847 A148848 A148849

Adjacent sequences: A163059 A163060 A163061 * A163063 A163064 A163065

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved