|
%I #40 Dec 03 2023 12:30:59
%S 2,1,9,13,49,101,297,701,1889,4693,12249,31021,80017,204101,524169,
%T 1340573,3437249,8799541,22548537,57746701,147940849,378927653,
%U 970691049,2486401661,6369165857,16314772501,41791435929,107050525933,274216269649,702418373381
%N Variant of Lucas numbers: a(n) = a(n-1) + 4*a(n-2) starting with a(0)=2 and a(1)=1.
%C 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
%C The Lucas sequence V(1,-4). - _Peter Bala_, Jun 23 2015
%D Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.
%H Alois P. Heinz, <a href="/A072265/b072265.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%F G.f.: (2-x)/(1-x-4*x^2). - _Gary W. Adamson_, Jul 02 2003
%F a(n) = ((1+sqrt(17))/2)^n + ((1-sqrt(17))/2)^n = 4*A006131(n-1) + A006131(n+1) = A075117(4, n).
%F a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 17*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F a(n) = 2^n * Lucas(n, 1/2). - _G. C. Greubel_, Jan 15 2020
%p a:= n-> (Matrix([[1,2]]). Matrix([[1,1], [4,0]])^n)[1,2]:
%p seq(a(n), n=0..32); # _Alois P. Heinz_, Aug 15 2008
%p a := n -> 2*(2*I)^n*ChebyshevT(n, -I/4):
%p seq(simplify(a(n)), n = 0..29); # _Peter Luschny_, Dec 03 2023
%t CoefficientList[Series[(2-x)/(1-x-4*x^2), {x,0,30}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)
%t Table[2^n*LucasL[n, 1/2], {n,0,30}] (* _G. C. Greubel_, Jan 15 2020 *)
%o (PARI) polsym(x^2-x-4, 44)
%o (Sage) [lucas_number2(n,1,-4) for n in range(0, 27)] # _Zerinvary Lajos_, Apr 30 2009
%o (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
%o (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
%Y Cf. A006131.
%K easy,nonn
%O 0,1
%A _Miklos Kristof_, Jul 08 2002
%E Edited and extended by _Henry Bottomley_, Sep 03 2002
|
