A048694 Generalized Pellian with second term equal to 7.

1, 7, 15, 37, 89, 215, 519, 1253, 3025, 7303, 17631, 42565, 102761, 248087, 598935, 1445957, 3490849, 8427655, 20346159, 49119973, 118586105, 286292183, 691170471, 1668633125, 4028436721, 9725506567

Offset: 0

Author

Barry E. Williams

Keywords

• easy
• nonn

Comments

Pisano period lengths: 1, 1, 8, 4, 12, 8, 6, 4, 24, 12, 24, 8, 28, 6, 24, 8, 16, 24, 40, 12, ... . - R. J. Mathar, Aug 10 2012

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (2,1)

Crossrefs

Cf. A001333, A000129, A048654, A048655.

Programs

Maple

with(combinat): a:=n->5*fibonacci(n, 2)+fibonacci(n+1, 2): seq(a(n), n=0..26); # Zerinvary Lajos, Apr 04 2008

Mathematica

a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{6},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{2,1},{1,7},40] (* Harvey P. Dale, Jul 22 2011 *)

Maxima

a[0]:1$
a[1]:7$
a[n]:=2*a[n-1]+a[n-2]$
A048694(n):=a[n]$
makelist(A048694(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

Formula

a(n) = ((6+sqrt(2))(1+sqrt(2))^n - (6-sqrt(2))(1-sqrt(2))^n)/2*sqrt(2).

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

G.f.: (1+5*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008

a(n) = ((1+sqrt(18))(1+sqrt(2))^n+(1-sqrt(18))(1-sqrt(2))^n)/2 offset 0. a(n) = first binomial transform of 1,6,2,12,4,24. - Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009