A048696 Generalized Pellian with second term equal to 9.

1, 9, 19, 47, 113, 273, 659, 1591, 3841, 9273, 22387, 54047, 130481, 315009, 760499, 1836007, 4432513, 10701033, 25834579, 62370191, 150574961, 363520113, 877615187, 2118750487, 5115116161, 12348982809, 29813081779, 71975146367

Offset: 0

Author

Barry E. Williams

Keywords

• easy
• nonn

Comments

Binomial transform of 5,6,10,12,20,24,40. - Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Binomial transform of A164587. Inverse binomial transform of A164298. - Klaus Brockhaus, Aug 17 2009

For n > 0: a(n) = A105082(n) - A105082(n-1). - Reinhard Zumkeller, Dec 15 2013

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (2,1).

Crossrefs

Cf. A001333, A000129, A048654, A048655.

Programs

Haskell

a048696 n = a048696_list !! n
a048696_list = 1 : 9 : zipWith (+)
               a048696_list (map (2 *) $ tail a048696_list)
-- Reinhard Zumkeller, Dec 15 2013

Magma

[ n le 2 select 8*n-7 else 2*Self(n-1)+Self(n-2): n in [1..28] ]; // Klaus Brockhaus, Aug 17 2009

Maple

with(combinat): a:=n->7*fibonacci(n, 2)+fibonacci(n+1, 2): seq(a(n), n=0..25); # Zerinvary Lajos, Apr 04 2008

Mathematica

a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{8},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{2,1},{1,9},30] (* Harvey P. Dale, Apr 20 2012 *)

Maxima

a[0]:1$
a[1]:9$
a[n]:=2*a[n-1]+a[n-2]$
A048696(n):=a[n]$
makelist(A048696(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

Formula

a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=9.

a(n) = ((4*sqrt(2)+1)(1+sqrt(2))^n - (4*sqrt(2)-1)(1-sqrt(2))^n)/2.

G.f.: (1+7*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008