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A134862 Wythoff ABB numbers.

%I A134862 #30 Mar 29 2025 12:16:03
%S A134862 8,21,29,42,55,63,76,84,97,110,118,131,144,152,165,173,186,199,207,
%T A134862 220,228,241,254,262,275,288,296,309,317,330,343,351,364,377,385,398,
%U A134862 406,419,432,440,453,461,474,487,495,508,521,529,542,550,563,576,584,597
%N A134862 Wythoff ABB numbers.
%C A134862 The lower and upper Wythoff sequences, A and B, satisfy the complementary equation ABB = 2A+3B.
%C A134862 The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - _Amiram Eldar_, Mar 24 2025
%H A134862 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. 5.
%H A134862 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.
%F A134862 a(n) = A(B(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.
%t A134862 A[n_] := Floor[n * GoldenRatio]; B[n_] := Floor[n * GoldenRatio^2]; a[n_] := A[B[B[n]]]; Array[a, 100] (* _Amiram Eldar_, Mar 24 2025 *)
%o A134862 (Python)
%o A134862 from sympy import floor
%o A134862 from mpmath import phi
%o A134862 def A(n): return floor(n*phi)
%o A134862 def B(n): return floor(n*phi**2)
%o A134862 def a(n): return A(B(B(n))) # _Indranil Ghosh_, Jun 10 2017
%o A134862 (Python)
%o A134862 from math import isqrt
%o A134862 def A134862(n): return 5*(n+isqrt(5*n**2)>>1)+3*n # _Chai Wah Wu_, Aug 10 2022
%Y A134862 Cf. A000201, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134860, A134861, A134863, A035338, A134864, A035513, A244593.
%Y A134862 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.
%K A134862 nonn
%O A134862 1,1
%A A134862 _Clark Kimberling_, Nov 14 2007