A163607 - OEIS

A163607

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

5, 9, 23, 55, 133, 321, 775, 1871, 4517, 10905, 26327, 63559, 153445, 370449, 894343, 2159135, 5212613, 12584361, 30381335, 73347031, 177075397, 427497825, 1032071047, 2491639919, 6015350885, 14522341689, 35060034263, 84642410215

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OFFSET

0,1

COMMENTS

Binomial transform of A163888. Inverse binomial transform of A163608.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1).

FORMULA

a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 5, a(1) = 9.

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

a(n) = 5*A000129(n+1) - A000129(n). - R. J. Mathar, Nov 08 2013

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

a(n) = 2*A001333(n) + A001333(n+2). - Philippe Deléham, Mar 06 2023

MATHEMATICA

LinearRecurrence[{2, 1}, {5, 9}, 50] (* G. C. Greubel, Jul 29 2017 *)

PROG

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)*(1+r)^n+(5-2*r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

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

CROSSREFS

Cf. A163608, A163888.

Cf. A000129, A001333.

Sequence in context: A280030 A074340 A079993 * A067294 A271543 A272256

Adjacent sequences: A163604 A163605 A163606 * A163608 A163609 A163610

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

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

STATUS

approved