A276863 First differences of the Beatty sequence A276854 for 1 + sqrt(5).

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

Offset: 1

Author

Clark Kimberling, Sep 24 2016

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A188187, A276854, A276881.

Programs

Mathematica

z = 500; r = 1+Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A276854 *)
Differences[b] (* A276863 *)

Python

from sympy import integer_nthroot
def A276863(n): return 1+integer_nthroot(5*n**2,2)[0]-integer_nthroot(5*(n-1)**2,2)[0] # Chai Wah Wu, Mar 16 2021

Formula

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

a(n) = A188187(n) + 3, as follows right from the definitions. - Michel Dekking, Sep 02 2019

a(n) = 1+floor(n*sqrt(5))-floor((n-1)*sqrt(5)). - Chai Wah Wu, Mar 16 2021