A003623 Wythoff AB-numbers: floor(floor(n*phi^2)*phi), where phi = (1+sqrt(5))/2. (Formerly M2715)

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

1,1

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

References

J. 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.

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, JIS 11 (2008) 08.3.3.

Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).

Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.

U. Larsson and N. Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.

F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015 (see page 2, essential numbers).

Index entries for sequences related to Beatty sequences

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).

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

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

Sequence in context: A047470 A184401 A190251 * A190463 A190435 A188032

Adjacent sequences: A003620 A003621 A003622 * A003624 A003625 A003626

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Name improved by Michel Dekking, Sep 07 2016

Status

approved