A143461 Square array A(n,k) of numbers of length n quaternary words with at least k 0-digits between any other digits (n,k >= 0), read by antidiagonals.

1, 1, 4, 1, 4, 16, 1, 4, 7, 64, 1, 4, 7, 19, 256, 1, 4, 7, 10, 40, 1024, 1, 4, 7, 10, 22, 97, 4096, 1, 4, 7, 10, 13, 43, 217, 16384, 1, 4, 7, 10, 13, 25, 73, 508, 65536, 1, 4, 7, 10, 13, 16, 46, 139, 1159, 262144, 1, 4, 7, 10, 13, 16, 28, 76, 268, 2683, 1048576, 1, 4, 7, 10, 13, 16, 19, 49, 115, 487, 6160, 4194304

Offset

0,3

Links

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

Formula

G.f. of column k: 1/(x^k*(1-x-3*x^(k+1))).

A(n,k) = 4^n if k=0, else A(n,k) = 3*n+1 if n<=k+1, else A(n,k) = A(n-1,k) + 3*A(n-k-1,k).

Example

A (3,1) = 19, because 19 quaternary words of length 3 have at least 1 0-digit between any other digits: 000, 001, 002, 003, 010, 020, 030, 100, 101, 102, 103, 200, 201, 202, 203, 300, 301, 301, 303.
Square array A(n,k) begins:
       1,   1,   1,  1,  1,  1,  1,  1,  ...
       4,   4,   4,  4,  4,  4,  4,  4,  ...
      16,   7,   7,  7,  7,  7,  7,  7,  ...
      64,  19,  10, 10, 10, 10, 10, 10,  ...
     256,  40,  22, 13, 13, 13, 13, 13,  ...
    1024,  97,  43, 25, 16, 16, 16, 16,  ...
    4096, 217,  73, 46, 28, 19, 19, 19,  ...
   16384, 508, 139, 76, 49, 31, 22, 22,  ...

Maple

A:= proc(n, k) option remember; if k=0 then 4^n elif n<=k+1 then 3*n+1 else A(n-1, k) +3*A(n-k-1, k) fi end: seq(seq(A(n, d-n), n=0..d), d=0..13);

Mathematica

a[n_, 0] := 4^n; a[n_, k_] /; n <= k+1 := 3*n+1; a[n_, k_] := a[n, k] = a[n-1, k] + 3*a[n-k-1, k]; Table[a[n-k, k], {n, 0, 13}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014, after Maple *)

Crossrefs

Columns k=0-9 give: A000302, A006130(n+1), A084386(n+2), A143454, A143455, A143456, A143457, A143458, A143459, A143460.

Main diagonal gives A016777.

Sequence in context: A228782 A205125 A248978 * A066808 A363935 A033918

Adjacent sequences: A143458 A143459 A143460 * A143462 A143463 A143464

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 16 2008

Status

approved