A276129 - OEIS

%I #35 Dec 27 2017 03:10:42

%S 1,3,6,13,27,54,106,204,387,725,1344,2469,4500,8145,14652,26213,46665,

%T 82704,145982,256722,449937,786109,1369494,2379447,4123944,7130895,

%U 12303714,21186013,36411399,62466906,106987282,182946888,312367887,532587461,906840060

%N 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.

%C 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.

%C 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) + ...

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

%C 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

%F 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

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

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

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

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

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

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

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

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

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

%e 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.

%p b:= proc(n, m) option remember;

%p       `if`(n+m=0, 1, `if`(m>0, b(n, m-1), 0)+

%p       add(`if`(j::odd, b(n-j, m), 0), j=1..n))

%p     end:

%p a:= n-> b(n, 2):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 29 2016

%t a[0] = 1; a[n_] := Sum[Binomial[n - 2*k + 2, 2]*Binomial[n - k - 1, k], {k, 0, (n - 1)/2}];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Dec 27 2017, after _Andrew Howroyd_ *)

%o (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

%Y Cf. A000045, A029907, A239242.

%K nonn

%O 0,2

%A _Gregory L. Simay_, Aug 21 2016

%E More terms from _Alois P. Heinz_, Aug 29 2016