%I #12 Oct 04 2018 20:23:30

%S 1,1,3,1,3,9,1,3,5,27,1,3,5,11,81,1,3,5,7,21,243,1,3,5,7,13,43,729,1,

%T 3,5,7,9,23,85,2187,1,3,5,7,9,15,37,171,6561,1,3,5,7,9,11,25,63,341,

%U 19683,1,3,5,7,9,11,17,39,109,683,59049,1,3,5,7,9,11,13,27,57,183,1365,177147

%N Square array A(n,k) of numbers of length n ternary words with at least k 0-digits between any other digits (n,k >= 0), read by antidiagonals.

%H Alois P. Heinz, <a href="/A143453/b143453.txt">Rows n = 0..140, flattened</a>

%F G.f. of column k: 1/(x^k*(1-x-2*x^(k+1))).

%F A(n,k) = 3^n if k=0, else A(n,k) = 2*n+1 if n<=k+1, else A(n,k) = A(n-1,k) + 2*A(n-k-1,k).

%e A(3,1) = 11, because 11 ternary words of length 3 have at least 1 0-digit between any other digits: 000, 001, 002, 010, 020, 100, 101, 102, 200, 201, 202.

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 3, 3, 3, 3, 3, 3, 3, 3, ...

%e 9, 5, 5, 5, 5, 5, 5, 5, ...

%e 27, 11, 7, 7, 7, 7, 7, 7, ...

%e 81, 21, 13, 9, 9, 9, 9, 9, ...

%e 243, 43, 23, 15, 11, 11, 11, 11, ...

%e 729, 85, 37, 25, 17, 13, 13, 13, ...

%e 2187, 171, 63, 39, 27, 19, 15, 15, ...

%p A := proc (n::nonnegint, k::nonnegint) option remember; if k=0 then 3^n elif n<=k+1 then 2*n+1 else A(n-1, k) +2*A(n-k-1, k) fi end: seq(seq(A(n,d-n), n=0..d), d=0..14);

%t a[n_, 0] := 3^n; a[n_, k_] /; n <= k+1 := 2*n+1; a[n_, k_] := a[n, k] = a[n-1, k] + 2*a[n-k-1, k]; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 11 2013 *)

%Y Column k=0: A000244, k=1: A001045(n+2), k=2: A003229(n+1) and A077949(n+2), k=3: A052942(n+3), k=4: A143447, k=5: A143448, k=6: A143449, k=7: A143450, k=8: A143451, k=9: A143452.

%Y Diagonal: A005408.

%K nonn,tabl

%O 0,3

%A _Alois P. Heinz_, Aug 16 2008