A050141 a(n) = 2*floor((n+1)*phi) - 2*floor(n*phi) - 1 where phi = (1 + sqrt(5))/2 is the golden ratio.

3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

First differences of A050140.

The substitutions 3->1 and 1->0 carry this sequence onto the infinite Fibonacci word (A005614).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Programs

Mathematica

#[[2]]-#[[1]]-1&/@Partition[Table[2*Floor[n*GoldenRatio],{n,100}],2,1] (* Harvey P. Dale, Mar 09 2025 *)

PARI

a(n)=(n+1+sqrtint(5*(n+1)^2))\2*2 - (n+sqrtint(5*n^2))\2*2 - 1 \\ Charles R Greathouse IV, Dec 28 2017

Python

from math import isqrt
def A050141(n): return -(n+isqrt(m:=5*n**2)&-2)+(n+1+isqrt(m+5*((n<<1)+1))&-2)-1
# Chai Wah Wu, May 04 2025

Extensions

Name edited by Michel Marcus and Charles R Greathouse IV, Dec 28 2017