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A246355 Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(2)..s(n+1) occurs for the k-th time.

%I A246355 #8 Sep 28 2014 11:08:55
%S A246355 2,5,2,7,5,2,10,7,7,2,13,10,10,7,2,15,13,15,10,7,2,18,15,20,15,10,10,
%T A246355 2,20,18,23,20,15,15,10,2,23,20,28,23,20,23,15,10,2,26,23,31,28,23,31,
%U A246355 23,15,10,2,28,26,36,31,28,36,31,23,15,10,2,31,28,41
%N A246355 Rectangular array:  T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(2)..s(n+1) occurs for the k-th time.
%C A246355 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 A246355 First 2 rows:  A001950 (upper Wythoff numbers);
%F A246355 next 3 rows:  A035336 (Wythoff BA numbers);
%F A246355 next 5 rows:  A134861 (Wythoff BAA numbers);
%F A246355 next 8 rows:  (Wythoff BAAA numbers).
%F A246355 (The patterns continue; in particular the number of identical consecutive rows is always a Fibonacci number, as in A000045.)
%e A246355 The upper Wythoff sequence, A001950 gives the positions of 1 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(2)..s(2) is simply 1, which occurs at positions 2,5,7,10,13,... as in row 1 of T.  For n = 5, the block s(2)..s(6) is 1,0,0,1,0 which occurs at positions 2,7,10,15,20,23, ...
%e A246355 The first 6 rows follow:
%e A246355 2 .. 5 .. 7 ... 10 .. 13 .. 15 .. 18 ...
%e A246355 2 .. 5 .. 7 ... 10 .. 13 .. 15 .. 18 ...
%e A246355 2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
%e A246355 2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
%e A246355 2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
%e A246355 2 .. 10 . 15 .. 23 .. 31 .. 36 .. 44 ...
%t A246355 z = 1000; s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, 12]]; Flatten[Position[s, 1]]; 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[2, 2 + k] &], z2]; Column[Table[t[k], {k, 0, z2}]](* A246355, array *)
%t A246355 w[n_, k_] := t[n][[k + 1]]; Table[w[n - k, k], {n, 0, z2 - 1}, {k, n, 0, -1}] // Flatten (* A246355, sequence *)
%Y A246355 Cf. A003849, A246354.
%K A246355 nonn,easy,tabl
%O A246355 1,1
%A A246355 _Clark Kimberling_, Aug 24 2014