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A193147 Expansion of 1/(1 - x - 2*x^3 - x^5).

%I A193147 #53 Jun 14 2024 10:31:17
%S A193147 1,1,1,3,5,8,15,26,45,80,140,245,431,756,1326,2328,4085,7168,12580,
%T A193147 22076,38740,67985,119305,209365,367411,644761,1131476,1985603,
%U A193147 3484490,6114853,10730820,18831276,33046585,57992715,101770120,178594110,313410816,549997641
%N A193147 Expansion of 1/(1 - x - 2*x^3 - x^5).
%C A193147 The Ze3 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.
%C A193147 Number of tilings of a 5 X 2n rectangle with 5 X 1 pentominoes. - _M. Poyraz Torcuk_, Dec 18 2021
%H A193147 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,0,1).
%F A193147 G.f.: 1/(1-x-2*x^3-x^5) = -1 / ( (1+x+x^2)*(x^3-x^2+2*x-1) ).
%F A193147 a(n) = a(n-1) + 2*a(n-3) + a(n-5) with a(n) = 0 for n= -4, -3, -2, -1 and a(0) = 1.
%F A193147 a(n) = (5*b(n+1) - 4*b(n) + 3*b(n-1) + 2*c(n) + 3*c(n-1))/7 with b(n) = A005314(n) and c(n) = A049347(n).
%F A193147 G.f.: 1 + x/(U(0)-x) where G(k)= 1 - x^2*(k+1)/(1 - 1/(1 + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 17 2012
%F A193147 a(n) = Sum_{m=floor((n+1)/2)..n} Sum_{j=0..2*m-n} C(j,3*n-5*m+2*j) * C(2*m-n,j) * 2^(3*n-5*m+2*j). - _Vladimir Kruchinin_, Mar 10 2013
%F A193147 With offset 1, the INVERT transform of (1 + 2x^2 + x^4). - _Gary W. Adamson_, Mar 30 2017
%F A193147 a(n) = Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k). - _Seiichi Manyama_, Jun 14 2024
%p A193147 A193147 := proc(n) option remember: if n>=-4 and n<=-1 then 0 elif n=0 then 1 else procname(n-1) + 2*procname(n-3) + procname(n-5) fi: end: seq(A193147(n), n=0..32);
%t A193147 Series[1/(1 - x - 2*x^3 - x^5), {x, 0, 32}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Apr 02 2015 *)
%o A193147 (Maxima)
%o A193147 a(n):=sum(sum(binomial(j,3*n-5*m+2*j)*binomial(2*m-n,j)*2^(3*n-5*m+2*j), j,0,2*m-n),m,floor((n+1)/2),n); /* _Vladimir Kruchinin_, Mar 10 2013 */
%Y A193147 Bisection of A003520.
%Y A193147 Cf. A035317, A180662, A373578.
%K A193147 nonn,easy
%O A193147 0,4
%A A193147 _Johannes W. Meijer_, Jul 20 2011