A072265 Variant of Lucas numbers: a(n) = a(n-1) + 4*a(n-2) starting with a(0)=2 and a(1)=1.

2, 1, 9, 13, 49, 101, 297, 701, 1889, 4693, 12249, 31021, 80017, 204101, 524169, 1340573, 3437249, 8799541, 22548537, 57746701, 147940849, 378927653, 970691049, 2486401661, 6369165857, 16314772501, 41791435929, 107050525933, 274216269649, 702418373381

Offset: 0

Author

Miklos Kristof, Jul 08 2002

Keywords

• easy
• nonn

Comments

Pisano period lengths: 1, 1, 8, 1, 6, 8, 48, 2, 24, 6,120, 8, 12, 48, 24, 4, 8, 24, 18, 6, ... . - R. J. Mathar, Aug 10 2012

The Lucas sequence V(1,-4). - Peter Bala, Jun 23 2015

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Lucas sequence
• Index entries for linear recurrences with constant coefficients, signature (1,4).

Crossrefs

Cf. A006131.

Programs

GAP

a:=[2,1];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020

Magma

I:=[2,1]; [n le 2 select I[n] else Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2020

Maple

a:= n-> (Matrix([[1,2]]). Matrix([[1,1], [4,0]])^n)[1,2]:
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 15 2008
a := n -> 2*(2*I)^n*ChebyshevT(n, -I/4):
seq(simplify(a(n)), n = 0..29);  # Peter Luschny, Dec 03 2023

Mathematica

CoefficientList[Series[(2-x)/(1-x-4*x^2), {x,0,30}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
Table[2^n*LucasL[n, 1/2], {n,0,30}] (* G. C. Greubel, Jan 15 2020 *)

PARI

polsym(x^2-x-4, 44)

Sage

[lucas_number2(n,1,-4) for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009

Formula

G.f.: (2-x)/(1-x-4*x^2). - Gary W. Adamson, Jul 02 2003

a(n) = ((1+sqrt(17))/2)^n + ((1-sqrt(17))/2)^n = 4*A006131(n-1) + A006131(n+1) = A075117(4, n).

a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 17*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

a(n) = 2^n * Lucas(n, 1/2). - G. C. Greubel, Jan 15 2020

Extensions

Edited and extended by Henry Bottomley, Sep 03 2002