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A134859 Wythoff AAA numbers.

%I A134859 #68 Jul 29 2025 15:35:55
%S A134859 1,6,9,14,19,22,27,30,35,40,43,48,53,56,61,64,69,74,77,82,85,90,95,98,
%T A134859 103,108,111,116,119,124,129,132,137,142,145,150,153,158,163,166,171,
%U A134859 174,179,184,187,192,197,200,205,208,213,218,221,226,229,234,239,242
%N A134859 Wythoff AAA numbers.
%C A134859 The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAA = AB - 2 and AAA = A + B - 2.
%C A134859 Also numbers with suffix string 001, when written in Zeckendorf representation (with leading zero for the first term). - _A.H.M. Smeets_, Mar 20 2024
%C A134859 The asymptotic density of this sequence is 1/phi^3 = phi^3 - 4 = A098317 - 4 = 0.236067... . - _Amiram Eldar_, Mar 24 2025
%H A134859 A.H.M. Smeets, <a href="/A134859/b134859.txt">Table of n, a(n) for n = 1..20000</a>
%H A134859 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 4.
%H A134859 Aviezri S. Fraenkel, <a href="http://dx.doi.org/10.1137/090758994">Complementary iterated floor words and the Flora game</a>, SIAM J. Discrete Math. 24 (2010), no. 2, 570-588.
%H A134859 Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Griffiths/gr48.html">On a Matrix Arising from a Family of Iterated Self-Compositions</a>, Journal of Integer Sequences, 18 (2015), Article 15.11.8.
%H A134859 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html">Complementary equations and Wythoff Sequences</a>, Journal of Integer Sequences, 11 (2008), Article 08.3.3.
%H A134859 Johan Kok, <a href="https://arxiv.org/abs/2507.16500">Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs</a>, arXiv:2507.16500 [math.CO], 2025. See pp. 5-6.
%F A134859 a(n) = A(A(A(n))), n >= 1, with A=A000201, the lower Wythoff sequence.
%F A134859 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
%F A134859 a(n) = A095098(n-1), n > 1. - _R. J. Mathar_, Oct 16 2009
%F A134859 From _A.H.M. Smeets_, Mar 23 2024: (Start)
%F A134859 a(n) = A(n) + B(n) - 2 (see Clark Kimberling 2008), with A=A000201, B=A001950, the lower and upper Wythoff sequences, respectively.
%F A134859 Equals {A003622}\{A134860} (= Wythoff AA \ Wythoff AAB). (End)
%e A134859 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.
%p A134859 # For Maple code for these Wythoff compound sequences see A003622. - _N. J. A. Sloane_, Mar 30 2016
%t A134859 A[n_] := Floor[n GoldenRatio];
%t A134859 a[n_] := A@ A@ A@ n;
%t A134859 a /@ Range[100] (* _Jean-François Alcover_, Oct 28 2019 *)
%o A134859 (Python)
%o A134859 from sympy import floor
%o A134859 from mpmath import phi
%o A134859 def A(n): return floor(n*phi)
%o A134859 def a(n): return A(A(A(n))) # _Indranil Ghosh_, Jun 10 2017
%o A134859 (Python)
%o A134859 from math import isqrt
%o A134859 def A134859(n): return ((n+isqrt(5*n**2)>>1)-1<<1)+n # _Chai Wah Wu_, Aug 10 2022
%Y A134859 Cf. A001622, A000201, A001950, A003622, A003623, A035336, A098317, A101864, A134860, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
%Y A134859 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.
%Y A134859 Essentially the same as A095098.
%K A134859 nonn
%O A134859 1,2
%A A134859 _Clark Kimberling_, Nov 14 2007
%E A134859 Incorrect PARI program removed by _R. J. Mathar_, Oct 16 2009