OFFSET
1,1
COMMENTS
Previous name was: "From a 3-way splitting of positive integers: [[n*phi^2]*phi]."
a(n) is odd if and only if n is odd. - Clark Kimberling, Apr 21 2011
A005614(a(n)-1)=1 and A005614(a(n))=1, n>=1. Because Wythoff AB-numbers (see the formula section) mark the first entry of pairs of 1s in the rabbit sequence A005614(n-1), n>=1. - Wolfdieter Lang, Jun 28 2011
a(n) = k if and only if A270788(k) = 3, where A270788 is the infinite Fibonacci word on {1,2,3}. - Michel Dekking, Sep 07 2016
The asymptotic density of this sequence is 1/phi^3 = phi^3 - 4 = A098317 - 4 = 0.236067... . - Amiram Eldar, Mar 24 2025
Positive integers k such that {k*phi} > 1 - {2*phi}, where {...} denotes the fractional part of its argument. This shows that the asymptotic density of the sequence is {2*phi} = 2*phi - 3 = sqrt(5)-2. This aligns with Eldar's observation since 2*phi - 3 = 1/phi^3. - Hans J. H. Tuenter, Feb 08 2026
REFERENCES
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..10000
J.-P. Allouche and F. M. 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. 3.
Benoit Cloitre and Jeffrey Shallit, Some Fibonacci-Related Sequences, arXiv:2312.11706 [math.CO], 2023.
Aviezri S. Fraenkel, The Raleigh game, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1.
Aviezri S. Fraenkel, Complementary iterated floor words and the Flora game, SIAM J. Discrete Math. 24 (2010), no. 2, 570-588. - From N. J. A. Sloane, May 06 2011
A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, Vol. 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.
Urban Larsson and Nathan Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, Vol. 18 (2015), Article 15.7.4.
F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2018 (see page 2, essential numbers).
FORMULA
MAPLE
A003623:=proc(n) return floor(floor(n*(3+sqrt(5))/2)*(1+sqrt(5))/2); end:seq(A003623(n), n=1..59); # Nathaniel Johnston, Apr 21 2011
MATHEMATICA
f[n_] := Floor[ GoldenRatio * Floor[ n * GoldenRatio^2]]; Array[f, 47]
(* another *) Table[n+2Floor[n*GoldenRatio], {n, 1, 100}]
PROG
(Python)
from sympy import floor
from mpmath import phi
def a(n): return floor(n*phi) + floor(n*phi**2) # Indranil Ghosh, Jun 10 2017
(Python)
from math import isqrt
def A003623(n): return (n+isqrt(5*n**2)&-2)+n # Chai Wah Wu, Aug 25 2022
(PARI) a(n)=(n+sqrtint(5*n^2))\2*2+n \\ Charles R Greathouse IV, Jan 25 2022
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Name improved by Michel Dekking, Sep 07 2016
STATUS
approved