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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 16 2008
STATUS
approved