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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A033918 A136467

Adjacent sequences:  A143458 A143459 A143460 * A143462 A143463 A143464

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 16 2008

STATUS

approved

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Last modified October 22 22:35 EDT 2021. Contains 348180 sequences. (Running on oeis4.)