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A048654 a(n) = 2a(n-1) + a(n-2); a(0)=1, a(1)=4. 29
1, 4, 9, 22, 53, 128, 309, 746, 1801, 4348, 10497, 25342, 61181, 147704, 356589, 860882, 2078353, 5017588, 12113529, 29244646, 70602821, 170450288, 411503397, 993457082, 2398417561, 5790292204 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generalized Pellian with second term equal to 4.

The generalized Pellian with second term equal to s has the terms a(n) = A000129(n)*s + A000129(n-1). The generating function is -(1+s*x-2*x)/(-1+2*x+x^2). - R. J. Mathar, Nov 22 2007

LINKS

T. D. Noe, Table of n, a(n) for n = 0..300

A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pps. 245-252.

A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (2,1)

FORMULA

a(n) = [ (3+sqrt(2))(1+sqrt(2))^n - (3-sqrt(2))(1-sqrt(2))^n ]/2*sqrt(2).

a(n) = 2*A000129(n+2) - 3*A000129(n+1). - Creighton Dement, Oct 27 2004

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

a(n) = binomial transform of 1, 3, 2, 6, 4, 12, ... . - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

E.g.f.: exp(x)*cosh(sqrt(2)*x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Vaclav Kotesovec, Feb 16 2015

MAPLE

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

MATHEMATICA

a=2; b=1; c=1; lst={b}; Do[c=a+b+c; AppendTo[lst, c]; a=b; b=c, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 23 2009 *)

LinearRecurrence[{2, 1}, {1, 4}, 30] (* Harvey P. Dale, Jul 27 2011 *)

PROG

(Haskell)

a048654 n = a048654_list !! n

a048654_list =

   1 : 4 : zipWith (+) a048654_list (map (* 2) $ tail a048654_list)

-- Reinhard Zumkeller, Aug 01 2011

(Maxima)

a[0]:1$

a[1]:4$

a[n]:=2*a[n-1]+a[n-2]$

A048654(n):=a[n]$

makelist(A048654(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

(PARI) a(n)=(([0, 1; 1, 2]^n)*[1, 4]~)[1] \\ Charles R Greathouse IV, May 18 2015

CROSSREFS

Cf. A001333, A000129, A048655, A038761, A084214, A100525.

Sequence in context: A032288 A076859 A042833 * A122626 A135025 A231213

Adjacent sequences:  A048651 A048652 A048653 * A048655 A048656 A048657

KEYWORD

easy,nice,nonn

AUTHOR

Barry E. Williams

STATUS

approved

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Last modified August 20 20:09 EDT 2017. Contains 290837 sequences.