A035337 Third column of Wythoff array.

3, 11, 16, 24, 32, 37, 45, 50, 58, 66, 71, 79, 87, 92, 100, 105, 113, 121, 126, 134, 139, 147, 155, 160, 168, 176, 181, 189, 194, 202, 210, 215, 223, 231, 236, 244, 249, 257, 265, 270, 278, 283, 291, 299, 304, 312

Offset: 0

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn

Comments

Also, positions of 3's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011

The formula a(n) = 3*A003622(n)-n+1 = 3AA(n)-n+1 conjectured by Layman below is correct, since it is well known that AA(n)+1 = B(n) = A(n)+n, where B = A001950, and so 3AA(n)-n+1 = 3B(n)-n-2 = 3A(n)+2n-2. - Michel Dekking, Aug 31 2017

From Amiram Eldar, Mar 21 2022: (Start)

Numbers k for which the Zeckendorf representation A014417(k) ends with 1, 0, 0.

The asymptotic density of this sequence is 1/phi^4 = 2/(7+3*sqrt(5)), where phi is the golden ratio (A001622). (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
• J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
• Clark Kimberling, Complementary equations and Wythoff Sequences, JIS, Vol. 11 (2008), Article 08.3.3.
• N. J. A. Sloane, Classic Sequences.

Crossrefs

Cf. A001622, A014417, A139764.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.

Programs

Maple

t := (1+sqrt(5))/2 ; [ seq(3*floor((n+1)*t)+2*n,n=0..80) ];

Mathematica

Table[3 Floor[n GoldenRatio] + 2 n - 2, {n, 46}] (* Michael De Vlieger, Aug 31 2017 *)

PARI

a(n) = 2*n + 3*floor((1+sqrt(5))*(n+1)/2); \\ Altug Alkan, Sep 18 2017

Python

from sympy import floor
from mpmath import phi
def a(n): return 3*floor((n + 1)*phi) + 2*n # Indranil Ghosh, Jun 10 2017

Python

from math import isqrt
def A035337(n): return 3*(n+isqrt(5*n**2)>>1)+(n-1<<1) # Chai Wah Wu, Aug 11 2022

Formula

a(n) = F(4)A(n)+F(3)(n-1) = 3A(n)+2n-2, where A = A000201 and F = A000045. - Michel Dekking, Aug 31 2017

It appears that a(n) = 3*A003622(n) - n + 1. - John W. Layman, Aug 25 2011