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%I #15 Feb 02 2018 09:04:16
%S 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,
%T 4096,1,4,7,10,13,43,217,16384,1,4,7,10,13,25,73,508,65536,1,4,7,10,
%U 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
%N 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.
%H Alois P. Heinz, <a href="/A143461/b143461.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: 1/(x^k*(1-x-3*x^(k+1))).
%F 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).
%e 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.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 4, 4, 4, 4, 4, 4, 4, 4, ...
%e 16, 7, 7, 7, 7, 7, 7, 7, ...
%e 64, 19, 10, 10, 10, 10, 10, 10, ...
%e 256, 40, 22, 13, 13, 13, 13, 13, ...
%e 1024, 97, 43, 25, 16, 16, 16, 16, ...
%e 4096, 217, 73, 46, 28, 19, 19, 19, ...
%e 16384, 508, 139, 76, 49, 31, 22, 22, ...
%p 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);
%t 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 *)
%Y Columns k=0-9 give: A000302, A006130(n+1), A084386(n+2), A143454, A143455, A143456, A143457, A143458, A143459, A143460.
%Y Main diagonal gives A016777.
%K nonn,tabl
%O 0,3
%A _Alois P. Heinz_, Aug 16 2008
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