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A349550 Meta-Wythoff array based on A097285: M = (M(n,k)), by downward antidiagonals; every row of M is eventually a row of the Wythoff array, W = A035513, and every row of W is a row of M; see Comments.

%I A349550 #8 Dec 15 2021 02:28:32
%S A349550 1,2,1,3,3,2,5,4,3,1,8,7,5,4,2,13,11,8,5,4,3,21,18,13,9,6,4,1,34,29,
%T A349550 21,14,10,7,5,2,55,47,34,23,16,11,6,5,3,89,76,55,37,26,18,11,7,5,4,
%U A349550 144,123,89,60,42,29,17,12,8,5,1,233,199,144,97,68,47
%N A349550 Meta-Wythoff array based on A097285: M = (M(n,k)), by downward antidiagonals; every row of M is eventually a row of the Wythoff array, W = A035513, and every row of W is a row of M; see Comments.
%C A349550 Suppose that (s(1), s(2), ...) is a sequence satisfying s(k) = s(k-1) + s(k-2) for k >= 3.  If s(1) and s(2) are positive integers, then there is an index n such that (s(n), s(n+1), ...) is a row of A035513. The n-th row of M is the sequence (s(1), s(2), ...), where (s(1), s(2)) are the n-th pair described in A097285.
%C A349550 Every row of W is a row of M; indeed, M consists of all tails of all rows of W.
%e A349550 Corner:
%e A349550   1, 2, 3,  5,  8, 13, 21, 34,  55,  89, 144, 233
%e A349550   1, 3, 4,  7, 11, 18, 29, 47,  76, 123, 199, 322
%e A349550   2, 3, 5,  8, 13, 21, 34, 55,  89, 144, 233, 377
%e A349550   1, 4, 5,  9, 14, 23, 37, 60,  97, 157, 254, 411
%e A349550   2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466
%e A349550   3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521
%e A349550   1, 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500
%e A349550   2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555
%e A349550   3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
%e A349550   4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665
%e A349550 Example: The first 7 pairs in A097285 are (1,2), (1,3), (2,3), (1,4), (2,4), (3,4), (1,5), so that the first 7 rows of M are
%e A349550 (1,2,3,5,8,...) = (row 1 of W) = Fibonacci numbers, A000045;
%e A349550 (1,3 4,7,11,...), which includes row 2 of W, the Lucas numbers, A000032;
%e A349550 (2,3,5,8,13,...), a tail of row 1 of W;
%e A349550 (1,4,5,9,14,...), which includes row 4 of W;
%e A349550 (2,4,6,10,16,...), which includes row 3 of W;
%e A349550 (3,4,7,11,18,...), which includes row 2 of W;
%e A349550 (1,5,6,11,17,...), which includes row 7 of W.
%t A349550 z1 = 30; zc = 20; zr = 20;
%t A349550 t1 = {1, 2}; Do[t1 = Join[t1, Riffle[Range[n - 1], n], {n}], {n, 3, z1}]; (* A097285 *)
%t A349550 t = Partition[t1, 2];
%t A349550 f[n_] := Fibonacci[n]; r = (1 + Sqrt[5])/2;
%t A349550 s[h_, k_] := Table[h*f[n - 1] + k*f[n], {n, 2, zc}];
%t A349550 w = Table[Join[{h = t[[n]][[1]], k = t[[n]][[2]]}, s[h, k]], {n, 1, zr}]
%t A349550 TableForm[w]  (* A349550 array *)
%t A349550 w1[n_, k_] := w[[n]][[k]];
%t A349550 Table[w1[n - k + 1, k], {n, 13}, {k, n, 1, -1}] // Flatten (* A349550 sequence *)
%Y A349550 Cf. A000032, A000045, A035513, A097285, A349551, A349552, A349553, A349554.
%K A349550 nonn,tabl
%O A349550 1,2
%A A349550 _Clark Kimberling_, Nov 21 2021