OFFSET
1,1
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
Positive integers k such that {k*phi} < {(k+3)*phi}, where phi = (1+sqrt(5))/2 and {...} denotes the fractional part. - Jeffrey Shallit, Sep 23 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.
FORMULA
a(n) = B(B(n)), n>=1, with B(k)=A001950(k) (Wythoff B-numbers). a(0)=0 with B(0)=0.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 28 2005
STATUS
approved