A276864 First differences of the Beatty sequence A001952 for 2 + sqrt(2).

3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3

Offset: 1

Author

Clark Kimberling, Sep 24 2016

Keywords

• nonn
• easy

Comments

Shifted by 1 (as one should) this is the unique fixed point of the morphism 3 -> 34, 4 -> 343. See A159684. - Michel Dekking, Aug 25 2019

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001952, A006337, A276882.

Programs

Magma

[Floor(n*(2 + Sqrt(2))) - Floor((n-1)*(2 + Sqrt(2))): n in [1..100]]; // G. C. Greubel, Aug 16 2018

Mathematica

z = 500; r = 2+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A001952 *)
Differences[b] (* A276864 *)

PARI

a(n) = 2 + sqrtint(2*n^2) - sqrtint(2*(n-1)^2) \\ Andrew Howroyd, Feb 15 2018

Formula

a(n) = floor(n*r) - floor(n*r - r), where r = 2 + sqrt(2), n >= 1.

a(n) = 2 + floor(n*sqrt(2)) - floor((n-1)*sqrt(2)). - Andrew Howroyd, Feb 15 2018

Extensions

Name corrected by Michel Dekking, Aug 25 2019