login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A134859 Wythoff AAA numbers. 14
1, 6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAA=AB-2 and AAA=A+B-2.

LINKS

Table of n, a(n) for n=1..58.

Aviezri S. Fraenkel, Complementary iterated floor words and the Flora game, SIAM J. Discrete Math. 24 (2010), no. 2, 570-588.

Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.

Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, 11 (2008) 08.3.3.

FORMULA

a(n) = A(A(A(n))), n>=1, with A=A000201, the lower Wythoff sequence.

a(n) = 2*floor(n*Phi^2)+n-2 where Phi=(1+sqrt(5))/2. - Benoit Cloitre, Apr 12 2008; R. J. Mathar, Oct 16 2009

a(n) = A095098(n-1), n>1. - R. J. Mathar, Oct 16 2009

EXAMPLE

Starting with A=(1,3,4,6,8,9,11,12,14,16,17,19,...), we have A(2)=3, so A(A(2))=4, so A(A(A(2)))=6.

MAPLE

# For Maple code for these Wythoff compound sequences see A003622. - N. J. A. Sloane, Mar 30 2016

MATHEMATICA

A[n_] := Floor[n GoldenRatio];

a[n_] := A@ A@ A@ n;

a /@ Range[100] (* Jean-Fran├žois Alcover, Oct 28 2019 *)

PROG

(Python)

from sympy import floor

from mpmath import phi

def A(n): return floor(n*phi)

def a(n): return A(A(A(n))) # Indranil Ghosh, Jun 10 2017

(Python)

from math import isqrt

def A134859(n): return ((n+isqrt(5*n**2)>>1)-1<<1)+n # Chai Wah Wu, Aug 10 2022

CROSSREFS

Cf. A001622, A000201, A001950, A003622, A003623, A035336, A101864, A134860, A035337, A134861, A134862, A134863, A035338, A134864, 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.

Sequence in context: A315986 A190461 A095098 * A315987 A315988 A315989

Adjacent sequences:  A134856 A134857 A134858 * A134860 A134861 A134862

KEYWORD

nonn,changed

AUTHOR

Clark Kimberling, Nov 14 2007

EXTENSIONS

Incorrect PARI program removed by R. J. Mathar, Oct 16 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)