A052532 - OEIS

%I #29 Sep 08 2022 08:44:59

%S 1,0,0,1,2,1,2,5,7,8,14,24,34,49,79,123,182,276,429,655,990,1513,2321,

%T 3537,5385,8229,12574,19175,29252,44670,68190,104043,158790,242398,

%U 369961,564604,861749,1315318,2007485,3063877,4676340,7137394,10893438

%N Expansion of (1-x)/(1-x-x^3-x^4+x^5).

%H Vincenzo Librandi, <a href="/A052532/b052532.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=462">Encyclopedia of Combinatorial Structures 462</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1,-1).

%F G.f.: (1 - x)/(1 - x - x^3 - x^4 + x^5).

%F a(n) = a(n-1) + a(n-3) + a(n-4) - a(n-5), with a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=2.

%F a(n) = Sum_{alpha = RootOf(1-x-x^3-x^4+x^5)} (1/8519)*(138 + 2003*alpha - 346*alpha^2 - 444*alpha^3 + 11*alpha^4)*alpha^(-1-n).

%p spec := [S,{S=Sequence(Prod(Z,Z,Z,Union(Z, Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t CoefficientList[Series[(1-x)/(1-x-x^3-x^4+x^5), {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 28 2014 *)

%t LinearRecurrence[{1,0,1,1,-1},{1,0,0,1,2},50] (* _Harvey P. Dale_, May 12 2018 *)

%o (PARI) my(x='x+O('x^50)); Vec((1-x)/(1-x-x^3-x^4+x^5)) \\ _G. C. Greubel_, May 09 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-x^3-x^4+x^5) )); // _G. C. Greubel_, May 09 2019

%o (Sage) ((1-x)/(1-x-x^3-x^4+x^5)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, May 09 2019

%o (GAP) a:=[1,0,0,1,2];; for n in [6..50] do a[n]:=a[n-1]+a[n-3]+a[n-4] -a[n-5]; od; a; # _G. C. Greubel_, May 09 2019

%K easy,nonn

%O 0,5

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _James A. Sellers_, Jun 05 2000