A048655 Generalized Pellian with second term equal to 5.

1, 5, 11, 27, 65, 157, 379, 915, 2209, 5333, 12875, 31083, 75041, 181165, 437371, 1055907, 2549185, 6154277, 14857739, 35869755, 86597249, 209064253, 504725755, 1218515763, 2941757281, 7102030325

Offset

0,2

Comments

Equals binomial transform of A143095: (1, 4, 2, 8, 4, 16, 8, 32, ...). - Gary W. Adamson, Jul 23 2008

Links

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

M. Bicknell, A primer on the Pell sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.

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.

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

Tanya Khovanova, Recursive sequences

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

Formula

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

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

a(n) = P(n) - 3*P(n+1) + 2*P(n+2). - Creighton Dement, Jan 18 2005

G.f.: (1+3*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008

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

a(n) = 3*Pell(n) + Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016

Maple

with(combinat): a:=n->3*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].{{4}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{2, 1}, {1, 5}, 30] (* Harvey P. Dale, Nov 05 2011 *)

Prog

(Maxima)
a[0]:1$
a[1]:5$
a[n]:=2*a[n-1]+a[n-2]$
A048655(n):=a[n]$
makelist(A048655(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
(PARI) a(n)=([0, 1; 1, 2]^n*[1; 5])[1, 1] \\ Charles R Greathouse IV, Feb 09 2017
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/(1-2*x-x^2))); // G. C. Greubel, Jul 26 2018

Crossrefs

Cf. A001333, A000129, A048654, A143095.

Sequence in context: A192300 A289775 A119503 * A181896 A041671 A215221

Adjacent sequences: A048652 A048653 A048654 * A048656 A048657 A048658

Keyword

easy,nice,nonn

Author

Barry E. Williams

Status

approved