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A134859 Wythoff AAA numbers. 13
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.

C. 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

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

CROSSREFS

Cf. 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: A129413 A190461 A095098 * A154778 A106350 A217851

Adjacent sequences:  A134856 A134857 A134858 * A134860 A134861 A134862

KEYWORD

nonn

AUTHOR

Clark Kimberling, Nov 14 2007

EXTENSIONS

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

STATUS

approved

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Last modified February 19 18:31 EST 2018. Contains 299356 sequences. (Running on oeis4.)