A162268 a(n) = ((5+sqrt(2))*(1+sqrt(2))^n + (5-sqrt(2))*(1-sqrt(2))^n)/2.

5, 7, 19, 45, 109, 263, 635, 1533, 3701, 8935, 21571, 52077, 125725, 303527, 732779, 1769085, 4270949, 10310983, 24892915, 60096813, 145086541, 350269895, 845626331, 2041522557, 4928671445, 11898865447, 28726402339, 69351670125

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A162396.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1)

Crossrefs

Cf. A162396.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(1+r)^n+(5-r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009

Mathematica

LinearRecurrence[{2,1}, {5,7}, 30] (* Vincenzo Librandi, Feb 03 2018 *)
Table[(4*LucasL[n, 2] + LucasL[n + 1, 2])/2, {n, 0, 30}] (* G. C. Greubel, Aug 17 2018 *)

PARI

x='x+O('x^30); Vec((5-3*x)/(1-2*x-x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

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

G.f.: (5-3*x)/(1-2*x-x^2).

a(n) = 5*A000129(n+1) - 3*A000129(n). - R. J. Mathar, Mar 06 2013

a(n) = 4*A001333(n) + A001333(n+1). - G. C. Greubel, Aug 17 2018

Extensions

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