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A072265
Variant of Lucas numbers: a(n) = a(n-1) + 4*a(n-2) starting with a(0)=2 and a(1)=1.
8
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
0,1
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.
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
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 *)
PROG
(PARI) polsym(x^2-x-4, 44)
(SageMath) [lucas_number2(n, 1, -4) for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009
(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
(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
CROSSREFS
Cf. A006131.
Sequence in context: A198204 A099599 A085488 * A192352 A180001 A204371
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof, Jul 08 2002
EXTENSIONS
Edited and extended by Henry Bottomley, Sep 03 2002
STATUS
approved