A048693 Generalized Pellian with 2nd term equal to 6.

1, 6, 13, 32, 77, 186, 449, 1084, 2617, 6318, 15253, 36824, 88901, 214626, 518153, 1250932, 3020017, 7290966, 17601949, 42494864, 102591677, 247678218, 597948113, 1443574444, 3485097001, 8413768446

Offset: 0

Author

Barry E. Williams

Keywords

• easy
• nonn

Comments

Pisano period lengths: 1, 2, 8, 4, 12, 8, 6, 8, 24, 12, 24, 8, 28, 6, 24, 16, 16, 24, 40, 12, ... (is this A175181?). - R. J. Mathar, Aug 10 2012

Examples

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

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->4*fibonacci(n-1,2)+fibonacci(n,2): seq(a(n), n=1..26); # Zerinvary Lajos, Apr 04 2008

Mathematica

a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{5},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{2,1},{1,6},30] (* Harvey P. Dale, Mar 29 2013 *)

Maxima

a[0]:1$
a[1]:6$
a[n]:=2*a[n-1]+a[n-2]$
A048693(n):=a[n]$
makelist(A048693(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

Formula

a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=6.

G.f.: (1+4*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008

a(n) = 4*A000129(n) + A000129(n+1). - R. J. Mathar, Aug 10 2012