This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A260311 #37 Mar 07 2025 01:29:53
%S A260311 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 A260311 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 A260311 5,3,5,3,2,3,5,3,5,5,3,5,3,5,5,3,5,3
%N A260311 Difference sequence of A260317.
%C A260311 Conjecture: a(n) is a Fibonacci number (A000045) for every n.
%C A260311 In fact, a(n) is in {1,2,3,5}; proved with the Walnut theorem-prover. - _Jeffrey Shallit_, Oct 12 2022
%C A260311 Comment from _Jonathan F. P. Grube_ and _Benjamin Mason_, Oct 10 2024 [Corrected by _Jonathan F. P. Grube_, Nov 05 2024]: (Start)
%C A260311 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_{i,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 A260311 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_{2k, j} = c_{2k+1, j} for all positive k and 0<j<n_{2k}. 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 Fibonacci sequence. (End)
%H A260311 Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, <a href="https://arxiv.org/abs/2503.04122">Using Walnut to solve problems from the OEIS</a>, arXiv:2503.04122 [math.NT], 2025.
%H A260311 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 A260311 r = GoldenRatio; z = 1060;
%t A260311 u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
%t A260311 s[m_, n_] := v[m] + v[n];
%t A260311 t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]; (* A259601 *)
%t A260311 w = Flatten[Table[Count[Flatten[t], n], {n, 1, z}]];
%t A260311 p0 = Flatten[Position[w, 0]] (* A260317 *)
%t A260311 d = Differences[p0] (* A260311 *)
%Y A260311 Cf. A259556, A259600, A259601, A260317, A000045.
%K A260311 nonn,easy
%O A260311 1,6
%A A260311 _Clark Kimberling_, Jul 31 2015