

A143453


Square array A(n,k) of numbers of length n ternary words with at least k 0digits 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
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OFFSET

0,3


LINKS

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


FORMULA

G.f. of column k: 1/(x^k*(1x2*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(n1,k) + 2*A(nk1,k).


EXAMPLE

A(3,1) = 11, because 11 ternary words of length 3 have at least 1 0digit 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(n1, k) +2*A(nk1, k) fi end: seq(seq(A(n, dn), 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[n1, k] + 2*a[nk1, k]; Table[a[nk, k], {n, 0, 14}, {k, n, 0, 1}] // Flatten (* JeanFrançois 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



