%I #30 Sep 08 2022 08:45:58
%S 1,1,2,2,4,5,8,10,15,20,29,39,55,75,105,144,200,275,381,525,726,1001,
%T 1383,1908,2635,3636,5020,6928,9564,13200,18221,25149,34714,47914,
%U 66136,91285,126000,173914,240051
%N Expansion of 1/(1 - x - x^2 + x^3 - x^4 + x^6).
%C The Gi2 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.
%H Vincenzo Librandi, <a href="/A193146/b193146.txt">Table of n, a(n) for n = 0..300</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,0,-1).
%F G.f.: 1/(1 - x - x^2 + x^3 - x^4 + x^6).
%F a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-6) with a(n) = 0 for n = -5, -4, -3, -2, -1 and a(0) = 1.
%F a(n) = b(n) + b(n-1) + b(n-3) - (1-(-1)^n)/2 with b(n) = A003269(n) and b(-3) = b(-2) = b(-1) = 0.
%F a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n-3*k/2)+1, n-2*k+1). - _Taras Goy_, Dec 24 2019
%p A193146 := proc(n) option remember: if n>=-5 and n<=-1 then 0 elif n=0 then 1 else procname(n-1) + procname(n-2) - procname(n-3) + procname(n-4) - procname(n-6) fi: end: seq(A193146(n), n=0..40);
%t CoefficientList[Series[1/(1-x-x^2+x^3-x^4+x^6), {x, 0, 40}], x] (* _Michael De Vlieger_, Dec 24 2019 *)
%t LinearRecurrence[{1,1,-1,1,0,-1},{1,1,2,2,4,5},50] (* _Harvey P. Dale_, Mar 27 2022 *)
%o Contribution by _Bruno Berselli_, Jul 22 2011: (Start)
%o (PARI) Vec(1/(1-x-x^2+x^3-x^4+x^6) +O(x^40)) /* show terms */
%o (Maxima) makelist(coeff(taylor(1/(1-x-x^2+x^3-x^4+x^6), x, 0, n), x, n), n, 0, 40);
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!(1/(1-x-x^2+x^3-x^4+x^6))); (End)
%o (Sage)
%o def A193146_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/(1-x-x^2+x^3-x^4+x^6) ).list()
%o A193146_list(40) # _G. C. Greubel_, Jan 01 2020
%o (GAP) a:=[1,1,2,2,4,5];; for n in [7..40] do a[n]:=a[n-1]+a[n-2]-a[n-3]+a[n-4]-a[n-6]; od; a; # _G. C. Greubel_, Jan 01 2020
%Y Cf. A003269.
%Y Cf. A035317, A180662.
%K nonn,easy
%O 0,3
%A _Johannes W. Meijer_, Jul 20 2011