A003623 Wythoff AB-numbers: floor(floor(n*phi^2)*phi), where phi = (1+sqrt(5))/2.

3, 8, 11, 16, 21, 24, 29, 32, 37, 42, 45, 50, 55, 58, 63, 66, 71, 76, 79, 84, 87, 92, 97, 100, 105, 110, 113, 118, 121, 126, 131, 134, 139, 144, 147, 152, 155, 160, 165, 168, 173, 176, 181, 186, 189, 194, 199, 202, 207, 210, 215, 220, 223, 228, 231, 236, 241, 244, 249

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice
• easy

Comments

Previous name was: "From a 3-way splitting of positive integers: [[n*phi^2]*phi]."

Union of A001950 & A003622 & A003623 = A000027.

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

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).
• Index entries for sequences related to Beatty sequences.

Crossrefs

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

Cf. A005614, A098317, A270788.

Programs

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}]

PARI

a(n)=(n+sqrtint(5*n^2))\2*2+n \\ Charles R Greathouse IV, Jan 25 2022

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

Formula

a(n) = floor(n*phi) + floor(n*phi^2) = A000201(n) + A001950(n).

a(n) = 2*floor(n*phi) + n = 2*A000201(n) + n.

a(n) = A(B(n)) with A(k):=A000201(k) and B(k):=A001950(k), k>=1 (Wythoff AB-numbers).

Extensions

Name improved by Michel Dekking, Sep 07 2016