A143453 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.

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

Offset: 0

Author

Alois P. Heinz, Aug 16 2008

Keywords

• nonn
• tabl

Examples

			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.
Square array A(n,k) begins:
     1,   1,  1,  1,  1,  1,  1,  1, ...
     3,   3,  3,  3,  3,  3,  3,  3, ...
     9,   5,  5,  5,  5,  5,  5,  5, ...
    27,  11,  7,  7,  7,  7,  7,  7, ...
    81,  21, 13,  9,  9,  9,  9,  9, ...
   243,  43, 23, 15, 11, 11, 11, 11, ...
   729,  85, 37, 25, 17, 13, 13, 13, ...
  2187, 171, 63, 39, 27, 19, 15, 15, ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

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.

Diagonal: A005408.

Programs

Maple

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);

Mathematica

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 *)

Formula

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

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).