A186007 Array by antidiagonals: R(i,j)=number of the row of the Wythoff array which includes row(i+j)-row(i).

1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 4, 1, 3, 2, 1, 1, 1, 2, 3, 1, 3, 1, 1, 1, 4, 1, 8, 3, 1, 2, 1, 3, 1, 3, 6, 1, 2, 4, 1, 3, 1, 1, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 1, 1, 5, 2, 11, 4, 1, 8

Offset

1,7

Comments

The rows of the Wythoff array are essentially the positive Fibonacci sequences. If i>=1 and j>=1, then row(i+j)-row(i) is a positive Fibonacci sequence and therefore a tail of a row of the Wythoff array.

Links

Clark Kimberling, Antidiagonals n=1..60, flattened

Example

Northwest corner:
1....1....1....2....1....3....2....1....4....3
1....1....1....3....1....4....2....1....6....3
1....3....1....2....1....3....8....1....4....11
1....1....1....3....1....3....2....1....4....3
1....1....2....3....1....4....2....1....6....3
Let W be the Wythoff array (A035513).
row 8 of W: 19,31,50,81,...
row 2 of W:  4,7,11,18,...
(row 8)-(row 2): 15,24,39,63,... a tail of row 4,
so that R(2,6)=4.

Mathematica

w[{n_, k_}] := w[{n, k}] = Fibonacci[k + 1] Floor[n GoldenRatio] + (n - 1) Fibonacci[k];
f = Map[w[{Plus @@ #, {1, 2}}] - w[{#[[1]], {1, 2}}] &, Flatten[Table[{k, z - k + 1}, {z, 15}, {k, z}], 1]];
Module[{n, z}, Table[n = 1; While[(z = 1; While[First[f[[k]]] >= w[{n, z}], z++]); f[[k]] != {w[{n, z - 1}], w[{n, z}]}, n++]; n, {k, 1, Length[f]}]] (* Peter J. C. Moses, Apr 13 2013 *)

Crossrefs

Cf. A035513, A185735 (addition table for Wythoff rows).

Sequence in context: A065371 A300982 A354203 * A212623 A229214 A349685

Adjacent sequences: A186004 A186005 A186006 * A186008 A186009 A186010

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 10 2011

Extensions

Corrections and additions by Clark Kimberling, Apr 13 2013

Status

approved