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

%I A255671 #16 Dec 12 2024 15:37:03
%S A255671 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,
%T A255671 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,
%U A255671 2,4,2,2,6,2,4,2,2,4,2,2,6,2,4,2,2,4
%N A255671 Number of the column of the Wythoff array (A035513) that contains U(n), where U = A001950, the upper Wythoff sequence.
%C A255671 All the terms are even, and every even positive integer occurs infinitely many times.
%C A255671 From _Michel Dekking_, Dec 09 2024 and Ad van Loon: (Start)
%C A255671 This sequence has a self-similarity property:
%C A255671       a(U(n)) = a(n) + 2 for all n.
%C A255671 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.
%C A255671 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).
%C A255671 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.
%C A255671 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)
%H A255671 Ad van Loon, <a href="https://fibonacciandi.com/the-structure-of-the-expansions/">The structure of the expansions</a>, See Section 5.
%F A255671 a(n) = 2 if and only if n = L(j) for some j; otherwise, n = U(k) for some k.
%F A255671 a(n) = A255670(n) + 1 = A035612(A001950(n)).
%e A255671 Corner of the Wythoff array:
%e A255671   1   2   3   5   8   13
%e A255671   4   7   11  18  29  47
%e A255671   6   10  16  26  42  68
%e A255671   9   15  24  39  63  102
%e A255671 L = (1,3,4,6,8,9,11,...); U = (2,5,7,10,13,15,18,...), so that
%e A255671 A255670 = (1,3,1,1,5,...) and A255671 = (2,4,2,2,6,...).
%t A255671 z = 13; r = GoldenRatio; f[1] = {1}; f[2] = {1, 2};
%t A255671 f[n_] := f[n] = Join[f[n - 1], Most[f[n - 2]], {n}]; f[z];
%t A255671 g[n_] := g[n] = f[z][[n]]; Table[g[n], {n, 1, 100}]  (* A035612 *)
%t A255671 Table[g[Floor[n*r]], {n, 1, (1/r) Length[f[z]]}]     (* A255670 *)
%t A255671 Table[g[Floor[n*r^2]], {n, 1, (1/r^2) Length[f[z]]}] (* A255671 *)
%Y A255671 Cf. A255670, A035612, A000201, A001950.
%K A255671 nonn,easy
%O A255671 1,1
%A A255671 _Clark Kimberling_, Mar 03 2015