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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A164308 A082511

Adjacent sequences:  A143450 A143451 A143452 * A143454 A143455 A143456

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 16 2008

STATUS

approved

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Last modified January 27 01:45 EST 2021. Contains 340443 sequences. (Running on oeis4.)