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

Author

Barry E. Williams

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A001076, A001077, A015448, A033887.

Programs

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 *)

PARI

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

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