A162270 a(n) = ((5+sqrt(2))*(3+sqrt(2))^n + (5-sqrt(2))*(3-sqrt(2))^n)/2.

5, 17, 67, 283, 1229, 5393, 23755, 104779, 462389, 2040881, 9008563, 39765211, 175531325, 774831473, 3420269563, 15097797067, 66644895461, 294184793297, 1298594491555, 5732273396251, 25303478936621, 111694959845969

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

• nonn
• easy

Comments

Third binomial transform of A162396.

Links

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

Crossrefs

Cf. A162396.

Programs

GAP

a:=[5,17];; for n in [3..25] do a[n]:=6*a[n-1]-7*a[n-2]; od; a; # Muniru A Asiru, Sep 28 2018

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(3+r)^n+(5-r)*(3-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009

Magma

I:=[5,17]; [n le 2 select I[n] else 6*Self(n-1) - 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 28 2018

Maple

seq(coeff(series((5-13*x)/(1-6*x+7*x^2),x,n+1), x, n), n = 0..25); # Muniru A Asiru, Sep 28 2018

Mathematica

LinearRecurrence[{6,-7},{5,17},30] (* Harvey P. Dale, Jun 04 2016 *)

PARI

x='x+O('x^30); Vec((5-13*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 28 2018

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 17.

G.f.: (5-13*x)/(1-6*x+7*x^2).

a(n) = 5*A081179(n+1)-13*A081179(n). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009