A120749 Numbers k such that {k* sqrt(2)} > 1/2, where { } = fractional part.
2, 4, 7, 9, 11, 12, 14, 16, 19, 21, 23, 24, 26, 28, 31, 33, 36, 38, 40, 41, 43, 45, 48, 50, 52, 53, 55, 57, 60, 62, 64, 65, 67, 69, 70, 72, 74, 77, 79, 81, 82, 84, 86, 89, 91, 93, 94, 96, 98, 101, 103, 106, 108, 110, 111, 113, 115, 118, 120, 122, 123, 125, 127, 130, 132, 134
Offset: 1
Keywords
Examples
Call the present sequence b and its complement a. Then
{r} = {1.4142...} = 0.4142... < 1/2, so a(1) = 1;
{2r} = 0.828... > 1/2, so b(1) = 2;
{3r} = 0.242... < 1/2, so a(2) = 3.
Links
- Clark Kimberling, Table of n, a(n) for n = 1..1000
- Henk Bruin and Robbert Fokkink, On the records and zeros of a deterministic random walk, arXiv:2503.11734 [math.DS], 2025. See pp. 1, 3-4.
Programs
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Mathematica
z = 150; r = Sqrt[2]; f[n_] := If[FractionalPart[n*r] < 1/2, 0, 1] Flatten[Position[Table[f[n], {n, 1, z}], 0]] (* A120243 *) Flatten[Position[Table[f[n], {n, 1, z}], 1]] (* A120749 *) Select[Range[200],FractionalPart[# Sqrt[2]]>1/2&] (* Harvey P. Dale, Aug 20 2024 *)
Extensions
Updated by Clark Kimberling, Sep 16 2014
Comments