A276866 First differences of the Beatty sequence A004976 for 2 + sqrt(5).

4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4

Offset: 1

Author

Clark Kimberling, Sep 24 2016

Keywords

• nonn
• easy

Comments

(a(n+1)) is the unique fixed point of the substitution 4 -> 4445, 5 -> 44454, since alpha = sqrt(5)-2 satisfies 1/(4+alpha) = alpha. See Allouche and Shallit on characteristic words. - Michel Dekking, Jan 30 2017

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 285.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004976, A276884.

Programs

Mathematica

z = 500; r = 2+Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A004976 *)
Differences[b] (* A276866 *)

Formula

a(n) = floor(n*r) - floor(n*r - r), where r = 2 + sqrt(5), n >= 1.