A134864 Wythoff BBB numbers.

13, 34, 47, 68, 89, 102, 123, 136, 157, 178, 191, 212, 233, 246, 267, 280, 301, 322, 335, 356, 369, 390, 411, 424, 445, 466, 479, 500, 513, 534, 555, 568, 589, 610, 623, 644, 657, 678, 699, 712, 733, 746, 767, 788, 801, 822, 843, 856, 877, 890, 911, 932, 945

Offset: 1

Author

Clark Kimberling, Nov 14 2007

Keywords

• nonn

Comments

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BBB = 3A+5B.

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Crossrefs

Cf. A000201, A001950, A003622, A003623, A035336, A094214, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134863, 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.

Programs

Maple

a:=n->floor(n*((1+sqrt(5))/2)^2): [a(a(a(n)))$n=1..55]; # Muniru A Asiru, Nov 24 2018

Mathematica

Nest[Quotient[#(3+Sqrt@5),2]&,#,3]&/@Range@100 (* Federico Provvedi, Nov 24 2018 *)
b[n_]:=Floor[n GoldenRatio^2]; a[n_]:=b[b[b[n]]]; Array[a, 60] (* Vincenzo Librandi, Nov 24 2018 *)

Python

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

Python

from math import isqrt
def A134864(n): return (m:=5*n)+(((n+isqrt(n*m))&-2)<<2) # Chai Wah Wu, Aug 10 2022

Formula

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