A163609 a(n) = ((5 + 2*sqrt(2))*(3 + sqrt(2))^n + (5 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.

5, 19, 79, 341, 1493, 6571, 28975, 127853, 564293, 2490787, 10994671, 48532517, 214232405, 945666811, 4174374031, 18426576509, 81338840837, 359047009459, 1584910170895, 6996131959157, 30882420558677, 136321599637963

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A163608. Third binomial transform of A163888. Inverse binomial transform of A163610.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A163608, A163610, A163888.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(3+r)^n+(5-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

Mathematica

LinearRecurrence[{6, -7}, {5, 19}, 50] (* G. C. Greubel, Jul 29 2017 *)

PARI

x='x+O('x^50); Vec((5-11*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 19.

G.f.: (5-11*x)/(1-6*x+7*x^2).

a(n) = 5*A081179(n+1) - 11*A081179(n). - R. J. Mathar, Nov 08 2013

E.g.f.: exp(3*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