|
%I #47 Sep 08 2022 08:44:59
%S 1,4,16,65,264,1072,4353,17676,71776,291457,1183504,4805792,19514625,
%T 79242004,321773808,1306609857,5305681432,21544499536,87484608001,
%U 355244113436,1442520953280,5857568421121,23785517797920,96584592144960,392195937000961
%N Expansion of 1/(1-4*x-x^3).
%C A transform of A000302 under the mapping mapping g(x) -> (1/(1-x^3)) * g(x/(1-x^3)). - _Paul Barry_, Oct 20 2004
%C a(n) equals the number of n-length words on {0,1,2,3,4} such that 0 appears only in a run which length is a multiple of 3. - _Milan Janjic_, Feb 17 2015
%H Vincenzo Librandi, <a href="/A052927/b052927.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=913">Encyclopedia of Combinatorial Structures 913</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,1).
%F G.f.: 1/(1-4*x-x^3).
%F a(n) = 4*a(n-1) + a(n-3), with a(0)=1, a(1)=4, a(2)=16.
%F a(n) = Sum_{r=RootOf(-1+4*z+z^3)} (1/283)*(64 + 9*r + 24*r^2)*r^(-1-n).
%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*4^(n-3*k). - _Paul Barry_, Oct 20 2004
%p spec:= [S,{S=Sequence(Union(Z,Z,Z,Z,Prod(Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p seq(coeff(series(1/(1-4*x-x^3), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Oct 17 2019
%t CoefficientList[Series[1/(1-4x-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,0,1},{1,4,16},40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2012 *)
%o (Magma) I:=[1, 4, 16]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 22 2012
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( 1/(1-4*x-x^3))); // _Marius A. Burtea_, Oct 18 2019
%o (PARI) my(x='x+O('x^30)); Vec(1/(1-4*x-x^3)) \\ _G. C. Greubel_, Oct 17 2019
%o (Sage)
%o def A052927_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/(1-4*x-x^3) ).list()
%o A052927_list(30) # _G. C. Greubel_, Oct 17 2019
%o (GAP) a:=[1,4,16];; for n in [4..30] do a[n]:=4*a[n-1]+a[n-3]; od; a; # _G. C. Greubel_, Oct 17 2019
%Y Cf. A099503.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
