%I #29 Oct 10 2024 15:20:09
%S 1,1,1,1,1,2,2,1,2,1,2,3,2,3,2,3,3,2,3,3,2,3,5,3,2,3,5,3,2,3,5,3,5,3,
%T 2,3,5,3,5,3,2,3,5,3,5,5,3,5,3,2,3,5,3,5,5,3,5,3,2,3,5,3,5,5,3,5,3,5,
%U 5,3,5,3,2,3,5,3,5,5,3,5,3,5,5,3,5,3
%N Difference sequence of A260317.
%C Conjecture: a(n) is a Fibonacci number (A000045) for every n.
%C In fact, a(n) is in {1,2,3,5}; proved with the Walnut theorem-prover. - _Jeffrey Shallit_, Oct 12 2022
%C Comment from _Jonathan F. P. Grube_ and _Benjamin Mason_, Oct 10 2024: (Start)
%C With an offset of 41, the sequence is of the form 2[c_{1,1},...,c_{1,n_1}]2[c_{2,1},...,c_{2,n_2}]2[c_{3,1},...,c_{3,n_3}]2..., where [c_{i,1},...,c_{1,n_i}] is the word 35(355)^{c_{i,1}}35(355)^{c_{i,2}}35...35(355)^{c_{i,n_i}}353 over the alphabet {3,5} for some nonnegative integers c_{i,j}. Furthermore c_{i,j} is in {1,2}. Proved with Walnut, the theorem-prover.
%C Conjectures: n_{2k-1} = n_{2k} and c_{2k-1, j} = c_{2k, j} for all positive k and 0<j<n_{2k}+1. Furthermore c_{i,j} = c_{i,n_i - j} for all positive i and 0<j<n_i + 1. Furthermore (n_{2k})_{k > 0} is the Fibonnaci sequence. (End)
%H The Walnut code at <a href="https://cs.uwaterloo.ca/~shallit/oeis-walnut.txt">https://cs.uwaterloo.ca/~shallit/oeis-walnut.txt</a> proves the conjecture. Walnut itself can be downloaded from <a href="https://cs.uwaterloo.ca/~shallit/walnut.html">https://cs.uwaterloo.ca/~shallit/walnut.html</a>.
%t r = GoldenRatio; z = 1060;
%t u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
%t s[m_, n_] := v[m] + v[n];
%t t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]; (* A259601 *)
%t w = Flatten[Table[Count[Flatten[t], n], {n, 1, z}]];
%t p0 = Flatten[Position[w, 0]] (* A260317 *)
%t d = Differences[p0] (* A260311 *)
%Y Cf. A259556, A259600, A259601, A260317, A000045.
%K nonn,easy
%O 1,6
%A _Clark Kimberling_, Jul 31 2015