A255670 Number of the column of the Wythoff array (A035513) that contains L(n), where L = A000201, the lower Wythoff sequence.

1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 7, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 9, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 7, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 7, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 5, 1, 3, 1, 1, 3

Offset: 1

Author

Clark Kimberling, Mar 03 2015

Keywords

• nonn
• easy

Comments

All the terms are odd, and every odd positive integer occurs infinitely many times.

Examples

			Corner of the Wythoff array:
  1   2   3   5   8   13
  4   7   11  18  29  47
  6   10  16  26  42  68
  9   15  24  39  63  102
L = (1,3,4,6,8,9,11,...); U = (2,5,7,10,13,15,18,...), so that
this sequence = (1,3,1,1,5,...) and A255671 = (2,4,2,2,6,...).
		

Links

• Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 39.

Crossrefs

Cf. A255671, A035612, A000201, A001950.

Programs

Mathematica

z = 13; r = GoldenRatio; f[1] = {1}; f[2] = {1, 2};
f[n_] := f[n] = Join[f[n - 1], Most[f[n - 2]], {n}]; f[z];
g[n_] := g[n] = f[z][[n]]; Table[g[n], {n, 1, 100}]  (* A035612 *)
Table[g[Floor[n*r]], {n, 1, (1/r) Length[f[z]]}]     (* A255670 *)
Table[g[Floor[n*r^2]], {n, 1, (1/r^2) Length[f[z]]}] (* A255671 *)

Formula

a(n) = A255671(n) - 1 = A035612(A000201(n)).

a(n) = 1 if and only if n = L(j) for some j; otherwise, n = U(k) for some k.