A101864 Wythoff BB numbers.

5, 13, 18, 26, 34, 39, 47, 52, 60, 68, 73, 81, 89, 94, 102, 107, 115, 123, 128, 136, 141, 149, 157, 162, 170, 178, 183, 191, 196, 204, 212, 217, 225, 233, 238, 246, 251, 259, 267, 272, 280, 285, 293, 301, 306, 314, 322, 327, 335, 340, 348, 356, 361, 369, 374, 382, 390, 395

Offset: 1

Author

N. J. A. Sloane, Jan 28 2005

Keywords

• nonn

Comments

a(n)-3 are also the positions of 1 in A188436. - Federico Provvedi, Nov 22 2018

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

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..2000
• Jean-Paul Allouche and F. Michel Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018-2019.
• 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, JIS 11 (2008), Article 08.3.3.
• Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik, Vol. 78, No. 2 (2021), pp. 1-8.
• Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, Vol. 123, No. 2 (2016), 267-273.

Crossrefs

Second row of A101858.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.

Cf. A094214, A188436.

Programs

Maple

b:=n->floor(n*((1+sqrt(5))/2)^2): seq(b(b(n)),n=1..60); # Muniru A Asiru, Dec 05 2018

Mathematica

b[n_] := Floor[n * GoldenRatio^2]; a[n_] := b[b[n]]; Array[a, 60] (* Amiram Eldar, Nov 22 2018 *)

Python

from sympy import S
for n in range(1,60): print(int(S.GoldenRatio**2*(int(n*S.GoldenRatio**2))), end=', ') # Stefano Spezia, Dec 06 2018

Formula

a(n) = B(B(n)), n>=1, with B(k)=A001950(k) (Wythoff B-numbers). a(0)=0 with B(0)=0.