A276129 a(n) is the number of ordered ways to tile a strip of length n+2 with white tiles of odd lengths summing to length n and two red squares.

1, 3, 6, 13, 27, 54, 106, 204, 387, 725, 1344, 2469, 4500, 8145, 14652, 26213, 46665, 82704, 145982, 256722, 449937, 786109, 1369494, 2379447, 4123944, 7130895, 12303714, 21186013, 36411399, 62466906, 106987282, 182946888, 312367887, 532587461, 906840060

Offset: 0

Author

Gregory L. Simay, Aug 21 2016

Keywords

• nonn

Comments

a(n) is a specific case of b(r,n), the number of ordered ways to rearrange a tiling of length n + r, with odd(1,3,5...) white tiles summing to n and r red squares.

Define the following summation: b(0,r,n) = b(r,n); b(1,r,n) = b(r, n-2) + b(r, n-4) + b(r, n-6) + ..; b(s, r, n) = b(s-1, r, n-2) + b(s-1, r, n-4) + b(r-1, s, n-6) + ...

The number of compositions of n with exactly r even numbers is b(r, r, n-2r).

Except for the initial 1, this is the p-INVERT transform of (1,0,1,0,1,0,...) for p(S) = (1 - S)^3. See A291219. - Clark Kimberling, Sep 04 2017

Examples

			Let 1,3 be the lengths of the odd tiles summing to 3 and let r,r be the two odd squares. Then the resulting number of compositions is a(3) = 13. The 6 compositions are 3,r,r; r,3,r; r,r,3; 1,1,1,r,r; 1,1,r,r,1; 1,r,r,1,1; r,r,1,1,1; 1,1,r,1,r; 1,r,1,r,1; r,1,r,1,1; r,1,1,r,1; 1,r,1,1,r; r,1,1,1,r.
		

Crossrefs

Cf. A000045, A029907, A239242.

Programs

Maple

b:= proc(n, m) option remember;
      `if`(n+m=0, 1, `if`(m>0, b(n, m-1), 0)+
      add(`if`(j::odd, b(n-j, m), 0), j=1..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 29 2016

Mathematica

a[0] = 1; a[n_] := Sum[Binomial[n - 2*k + 2, 2]*Binomial[n - k - 1, k], {k, 0, (n - 1)/2}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 27 2017, after Andrew Howroyd *)

PARI

a(n)={if(n<=0, n==0, sum(k=0, (n-1)\2, binomial(n-2*k+2, 2)*binomial(n-k-1, k)))} \\ Andrew Howroyd, Dec 26 2017

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-2*k+2, 2)*binomial(n-k-1, k) for n > 0. - Andrew Howroyd, Dec 26 2017

b(0,0)=1. For n>=1, b(0,n) = b(0,0,n) = the n-th Fibonacci number, A000045(n).

b(1,0)=1. For n>=1, b(1,n) = A239342(n+1).

b(2,n) = a(n) = a(n-1) + a(n-2) + A239342(n+1) + A239342(n-1).

G.f. for b(2,n): ((1-x^2)/(1-x-x^2))^3.

G.f. for b(r,n): ((1-x^2)/(1-x-x^2))^(r+1).

b(1,1,n) = A029907(n+1).

b(r,n) = b(r, n-1) + b(r, n-2) + b(r-1, n) - b(r-1, n-2).

b(r,r,n) = b(r-1, r-1, n) + b(r, r, n-1) + b(r, r, n-2).

G.f. for b(r,r,n): (1-x^2)/((1-x-x^2)^(r+1)).

Extensions

More terms from Alois P. Heinz, Aug 29 2016