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

0,3

Links

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

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).

Example

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, ...

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 *)

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.

Sequence in context: A289067 A010282 A119265 * A248830 A350562 A164308

Adjacent sequences: A143450 A143451 A143452 * A143454 A143455 A143456

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 16 2008

Status

approved