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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.
1
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, 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, 144, 123, 89, 60, 42, 29, 17, 12, 8, 5, 1, 233, 199, 144, 97, 68, 47
OFFSET
1,2
COMMENTS
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.
Every row of W is a row of M; indeed, M consists of all tails of all rows of W.
EXAMPLE
Corner:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322
2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377
1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411
2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466
3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521
1, 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500
2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555
3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665
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
(1,2,3,5,8,...) = (row 1 of W) = Fibonacci numbers, A000045;
(1,3 4,7,11,...), which includes row 2 of W, the Lucas numbers, A000032;
(2,3,5,8,13,...), a tail of row 1 of W;
(1,4,5,9,14,...), which includes row 4 of W;
(2,4,6,10,16,...), which includes row 3 of W;
(3,4,7,11,18,...), which includes row 2 of W;
(1,5,6,11,17,...), which includes row 7 of W.
MATHEMATICA
z1 = 30; zc = 20; zr = 20;
t1 = {1, 2}; Do[t1 = Join[t1, Riffle[Range[n - 1], n], {n}], {n, 3, z1}]; (* A097285 *)
t = Partition[t1, 2];
f[n_] := Fibonacci[n]; r = (1 + Sqrt[5])/2;
s[h_, k_] := Table[h*f[n - 1] + k*f[n], {n, 2, zc}];
w = Table[Join[{h = t[[n]][[1]], k = t[[n]][[2]]}, s[h, k]], {n, 1, zr}]
TableForm[w] (* A349550 array *)
w1[n_, k_] := w[[n]][[k]];
Table[w1[n - k + 1, k], {n, 13}, {k, n, 1, -1}] // Flatten (* A349550 sequence *)
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 21 2021
STATUS
approved