A134859 Wythoff AAA numbers.

1, 6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242

Offset: 1

Author

Clark Kimberling, Nov 14 2007

Keywords

• nonn

Comments

The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAA = AB - 2 and AAA = A + B - 2.

Also numbers with suffix string 001, when written in Zeckendorf representation (with leading zero for the first term). - A.H.M. Smeets, Mar 20 2024

The asymptotic density of this sequence is 1/phi^3 = phi^3 - 4 = A098317 - 4 = 0.236067... . - Amiram Eldar, Mar 24 2025

Examples

			Starting with A=(1,3,4,6,8,9,11,12,14,16,17,19,...), we have A(2)=3, so A(A(2))=4, so A(A(A(2)))=6.
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..20000
• 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.
• Aviezri S. Fraenkel, Complementary iterated floor words and the Flora game, SIAM J. Discrete Math. 24 (2010), no. 2, 570-588.
• Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), Article 15.11.8.
• Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, 11 (2008), Article 08.3.3.
• Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See pp. 5-6.

Crossrefs

Cf. A001622, A000201, A001950, A003622, A003623, A035336, A098317, A101864, A134860, A035337, A134861, A134862, A134863, A035338, A134864, A035513.

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.

Essentially the same as A095098.

Programs

Maple

# For Maple code for these Wythoff compound sequences see A003622. - N. J. A. Sloane, Mar 30 2016

Mathematica

A[n_] := Floor[n GoldenRatio];
a[n_] := A@ A@ A@ n;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019 *)

Python

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

Python

from math import isqrt
def A134859(n): return ((n+isqrt(5*n**2)>>1)-1<<1)+n # Chai Wah Wu, Aug 10 2022

Formula

a(n) = A(A(A(n))), n >= 1, with A=A000201, the lower Wythoff sequence.

a(n) = 2*floor(n*Phi^2) - n - 2 where Phi = (1+sqrt(5))/2. - Benoit Cloitre, Apr 12 2008; R. J. Mathar, Oct 16 2009

a(n) = A095098(n-1), n > 1. - R. J. Mathar, Oct 16 2009

From A.H.M. Smeets, Mar 23 2024: (Start)

a(n) = A(n) + B(n) - 2 (see Clark Kimberling 2008), with A=A000201, B=A001950, the lower and upper Wythoff sequences, respectively.

Equals {A003622}\{A134860} (= Wythoff AA \ Wythoff AAB). (End)

Extensions

Incorrect PARI program removed by R. J. Mathar, Oct 16 2009