cp's OEIS Frontend

A260317 Numbers not of the form v(m) + v(n), where v = A001950 (upper Wythoff numbers) and 1 <= m <= n - 1, for n >= 2.

%I A260317 #20 Mar 07 2025 09:06:10
%S A260317 1,2,3,4,5,6,8,10,11,13,14,16,19,21,24,26,29,32,34,37,40,42,45,50,53,
%T A260317 55,58,63,66,68,71,76,79,84,87,89,92,97,100,105,108,110,113,118,121,
%U A260317 126,131,134,139,142,144,147,152,155,160,165,168,173,176,178,181
%N A260317 Numbers not of the form v(m) + v(n), where v = A001950 (upper Wythoff numbers) and 1 <= m <= n - 1, for n >= 2.
%C A260317 It appears that the difference sequence consists entirely of Fibonacci numbers (A000045); see A260311.
%C A260317 In fact, the difference sequence consists only of the numbers 1,2,3,5.  Proved with the Walnut theorem-prover. - _Jeffrey Shallit_, Oct 12 2022
%H A260317 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. See pp. 2-3.
%F A260317 n <= a(n) < 5n, see Shallit comment. - _Charles R Greathouse IV_, Nov 22 2024
%t A260317 r = GoldenRatio; z = 1060;
%t A260317 u[n_] := u[n] = Floor[n*r]; v[n_] := v[n] = Floor[n*r^2];
%t A260317 s[m_, n_] := v[m] + v[n];
%t A260317 t = Table[s[m, n], {n, 2, z}, {m, 1, n - 1}]; (* A259601 *)
%t A260317 w = Flatten[Table[Count[Flatten[t], n], {n, 1, z}]];
%t A260317 p0 = Flatten[Position[w, 0]]  (* A260317 *)
%t A260317 d = Differences[p0] (* A260311 *)
%Y A260317 Cf. A000045, A259556, A259600, A259601, A260311.
%K A260317 nonn,easy
%O A260317 1,2
%A A260317 _Clark Kimberling_, Jul 22 2015