A134861 Wythoff BAA numbers.

2, 10, 15, 23, 31, 36, 44, 49, 57, 65, 70, 78, 86, 91, 99, 104, 112, 120, 125, 133, 138, 146, 154, 159, 167, 175, 180, 188, 193, 201, 209, 214, 222, 230, 235, 243, 248, 256, 264, 269, 277, 282, 290, 298, 303, 311, 319, 324, 332, 337, 345, 353, 358, 366, 371

Offset

1,1

Comments

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BAA = A+2B-3.

Also numbers with suffix string 0010, when written in Zeckendorf representation (with leading zero's for the first term). - A.H.M. Smeets, Mar 20 2024

The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

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.

Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Formula

a(n) = B(A(A(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.

Mathematica

A[n_] := Floor[n * GoldenRatio]; B[n_] := Floor[n * GoldenRatio^2]; a[n_] := B[A[A[n]]]; Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

Prog

(Python)
from sympy import floor
from mpmath import phi
def A(n): return floor(n*phi)
def B(n): return floor(n*phi**2)
def a(n): return B(A(A(n))) # Indranil Ghosh, Jun 10 2017
(Python)
from math import isqrt
def A134861(n): return 3*((n+isqrt(5*n**2)>>1)-1)+(n<<1) # Chai Wah Wu, Aug 10 2022

Crossrefs

Cf. A000201, A001950, A003622, A003623, A035336, A094214, A101864, A134859, A035337, A134860, 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.

Sequence in context: A031222 A272041 A212160 * A063610 A392601 A391864

Adjacent sequences: A134858 A134859 A134860 * A134862 A134863 A134864

Keyword

nonn

Author

Clark Kimberling, Nov 14 2007

Status

approved