A048876 a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=7.

1, 7, 29, 123, 521, 2207, 9349, 39603, 167761, 710647, 3010349, 12752043, 54018521, 228826127, 969323029, 4106118243, 17393796001, 73681302247, 312119004989, 1322157322203, 5600748293801, 23725150497407, 100501350283429, 425730551631123, 1803423556807921

Offset

0,2

Comments

Generalized Pell equation with second term of 7.

Links

T. D. Noe, Table of n, a(n) for n = 0..200

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

L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.

Tanya Khovanova, Recursive Sequences

A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.

Index entries for linear recurrences with constant coefficients, signature (4,1).

Formula

G.f.: (1+3*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008

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

a(n) = A000032(3*n+1). - Thomas Baruchel, Nov 26 2003

From Gary Detlefs, Mar 06 2011: (Start)

a(n) = Fibonacci(3*n+7) mod Fibonacci(3*n+3), n > 0.

a(n) = Fibonacci(3*n+3) - Fibonacci(3*n-1). (End)

a(n) = A001076(n+1)+3*A001076(n). - R. J. Mathar, Oct 22 2013

a(n) = 5*F(2*n)*F(n+1) - L(n-1)*(-1)^n. - J. M. Bergot, Mar 22 2016

a(n) = Sum_{k=0..n} binomial(n,k)*5^floor((k+1)/2)*2^(n-k). - Tony Foster III, Sep 03 2017

Maple

with(combinat): a:=n->3*fibonacci(n-1, 4)+fibonacci(n, 4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

Mathematica

f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(2 + s)^n + (1 - s)(2 - s)^n)/2]]; (* Or *)
f[n_] := Fibonacci[3 n + 3] - Fibonacci[3 n - 1]; (* Or *)
f[n_] := Mod[ Fibonacci[3n + 7], Fibonacci[3n + 3]]; Array[f, 22, 0]
a[n_] := 4a[n - 1] + a[n - 2]; a[0] = 1; a[1] = 7; Array[a, 22, 0] (* Or *)
CoefficientList[ Series[(1 + 3x)/(1 - 4x - x^2), {x, 0, 21}], x] (* Robert G. Wilson v *)
LinearRecurrence[{4, 1}, {1, 7}, 30] (* Harvey P. Dale, Jun 13 2015 *)
Table[LucasL[3*n + 1], {n, 0, 20}] (* Rigoberto Florez, Apr 04 2019 *)

Prog

(PARI) Vec((1+3*x)/(1-4*x-x^2) + O(x^30)) \\ Altug Alkan, Oct 07 2015

Crossrefs

Cf. A001076, A001077, A015448, A033887.

Sequence in context: A037576 A327587 A055427 * A126394 A252832 A074468

Adjacent sequences: A048873 A048874 A048875 * A048877 A048878 A048879

Keyword

easy,nonn

Author

Barry E. Williams

Status

approved