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 A120243 #14 Mar 24 2025 12:47:37
%S A120243 1,3,5,6,8,10,13,15,17,18,20,22,25,27,29,30,32,34,35,37,39,42,44,46,
%T A120243 47,49,51,54,56,58,59,61,63,66,68,71,73,75,76,78,80,83,85,87,88,90,92,
%U A120243 95,97,99,100,102,104,105,107,109,112,114,116,117,119,121,124,126,128,129
%N A120243 Numbers k such that {k*sqrt(2)} < 1/2, where { } = fractional part.
%C A120243 The complement of a is b=A120749. Is a(n) < b(n) for all n? If k is a positive integer, then is b(n) - a(n) = k for infinitely many n?
%H A120243 Clark Kimberling, <a href="/A120243/b120243.txt">Table of n, a(n) for n = 1..1000</a>
%H A120243 Henk Bruin and Robbert Fokkink, <a href="https://arxiv.org/abs/2503.11734">On the records and zeros of a deterministic random walk</a>, arXiv:2503.11734 [math.DS], 2025. See pp. 1, 4.
%e A120243 {r} = {1.4142...} = 0.4142... < 1/2, so a(1)=1.
%e A120243 {2r} = 0.828... > 1/2, so b(1) = 2, where b = complement of a.
%e A120243 {3r} = 0.242... < 1/2, so a(2) = 3.
%t A120243 z = 150; r = Sqrt[2]; f[n_] := If[FractionalPart[n*r] < 1/2, 0, 1]
%t A120243 Flatten[Position[Table[f[n], {n, 1, z}], 0]] (* A120243 *)
%t A120243 Flatten[Position[Table[f[n], {n, 1, z}], 1]] (* A120749 *)
%Y A120243 Cf. A120749, A120750, A120751.
%K A120243 nonn
%O A120243 1,2
%A A120243 _Clark Kimberling_, Jul 01 2006
%E A120243 Updated by _Clark Kimberling_, Sep 16 2014