A255671 Number of the column of the Wythoff array (A035513) that contains U(n), where U = A001950, the upper Wythoff sequence.

2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 10, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4

Offset: 1

Author

Clark Kimberling, Mar 03 2015

Keywords

• nonn
• easy

Comments

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

From Michel Dekking, Dec 09 2024 and Ad van Loon: (Start)

This sequence has a self-similarity property:

a(U(n)) = a(n) + 2 for all n.

Proof: it is known that the columns C_h of the Wythoff array are compound Wythoff sequences. For example: C_1 = L^2, C_2 = UL.

In general column C_h is equal to LU^{(h-1)/2} if h is odd, and to U^{h/2}L if h is even (see Theorem 10 in Kimberling’s 2008 paper in JIS).

Now if h is odd then the elements of column C_h are a subsequence of L, so no U(m) can occur in such a column.

If h is even then the elements of column C_h form a subsequence of U, and so many U(m) occur. Suppose that a(m) = h. Then U(U(m)) is an element of column UU^{h/2}L = U^{(h+2)/2}L. This implies a(U(m)) = a(m) +2. (End)

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
A255670 = (1,3,1,1,5,...) and A255671 = (2,4,2,2,6,...).
		

Links

• Ad van Loon, The structure of the expansions, See Section 5.

Crossrefs

Cf. A255670, 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) = 2 if and only if n = L(j) for some j; otherwise, n = U(k) for some k.

a(n) = A255670(n) + 1 = A035612(A001950(n)).