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%I #9 Sep 28 2014 11:25:56
%S 1,3,1,4,4,1,6,6,4,1,8,9,6,6,1,9,12,9,9,6,1,11,14,12,14,9,6,1,12,17,
%T 14,19,14,9,9,1,14,19,17,22,19,14,14,9,1,16,22,19,27,22,19,22,14,9,1,
%U 17,25,22,30,27,22,30,22,14,9,1,19,27,25,35,30,27,35
%N Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(1)..s(n) occurs for the k-th time.
%C Assuming that every row of T is infinite, each row contains the next row as a proper subsequence. Row 1 of A246354 and row 1 of A246355 partition the positive integers.
%F First row: A000201 (lower Wythoff numbers);
%F next 2 rows: A003622 (Wythoff AA numbers);
%F next 3 rows: A134859 (Wythoff AAA numbers);
%F next 5 rows: A151915 (Wythoff AAAA numbers).
%F (The patterns continue; in particular the number of identical consecutive rows is always a Fibonacci number, as in A000045.)
%e The lower Wythoff sequence, A000201 gives the positions of 0 in A003849, which begins thus: 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1. For n = 1, the block s(1)..s(1) is simply 0, which occurs at positions 1,3,4,6,8,... as in row 1 of T. For n = 5, the block s(1)..s(5) is 0,1,0,0,1, which occurs at positions 1,6,9,14,19, ...
%e The first 7 rows follow:
%e 1 .. 3 .. 4 ... 6 ... 8 ... 9 ... 11 .. 12 ...
%e 1 .. 4 .. 6 ... 9 ... 12 .. 14 .. 17 .. 19 ...
%e 1 .. 4 .. 6 ... 9 ... 12 .. 14 .. 17 .. 19 ...
%e 1 .. 6 .. 9 ... 14 .. 19 .. 22 .. 27 .. 30 ...
%e 1 .. 6 .. 9 ... 14 .. 19 .. 22 .. 27 .. 30 ...
%e 1 .. 6 .. 9 ... 14 .. 19 .. 22 .. 27 .. 30 ...
%e 1 .. 9 .. 14 .. 22 .. 30 .. 35 .. 43 .. 48 ...
%t z = 1000; s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, 12]]; Flatten[Position[s, 0]]; b[m_, n_] := b[m, n] = Take[s, {m, n}]; z1 = 500; z2 = 12; t[k_] := t[k] = Take[Select[Range[1, z1], b[#, # + k] == b[1, 1 + k] &], z2]; Column[Table[t[k], {k, 0, z2}]] (* A246354, array *)
%t w[n_, k_] := t[n][[k + 1]]; Table[w[n - k, k], {n, 0, z2 - 1}, {k, n, 0, -1}] // Flatten (* A246354, sequence *)
%Y Cf. A003849, A246355.
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Aug 24 2014