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

1, 1, 1, 3, 5, 8, 15, 26, 45, 80, 140, 245, 431, 756, 1326, 2328, 4085, 7168, 12580, 22076, 38740, 67985, 119305, 209365, 367411, 644761, 1131476, 1985603, 3484490, 6114853, 10730820, 18831276, 33046585, 57992715, 101770120, 178594110, 313410816, 549997641

Offset: 0

Author

Johannes W. Meijer, Jul 20 2011

Keywords

• nonn
• easy

Comments

The Ze3 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.

Number of tilings of a 5 X 2n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Dec 18 2021

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,1).

Crossrefs

Bisection of A003520.

Cf. A035317, A180662, A373578.

Programs

Maple

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);

Mathematica

Series[1/(1 - x - 2*x^3 - x^5), {x, 0, 32}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2015 *)

Maxima

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 */

Formula

G.f.: 1/(1-x-2*x^3-x^5) = -1 / ( (1+x+x^2)*(x^3-x^2+2*x-1) ).

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.

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).

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

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

With offset 1, the INVERT transform of (1 + 2x^2 + x^4). - Gary W. Adamson, Mar 30 2017

a(n) = Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k). - Seiichi Manyama, Jun 14 2024