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A134864 Wythoff BBB numbers.

%I A134864 #33 Mar 24 2025 06:01:57
%S A134864 13,34,47,68,89,102,123,136,157,178,191,212,233,246,267,280,301,322,
%T A134864 335,356,369,390,411,424,445,466,479,500,513,534,555,568,589,610,623,
%U A134864 644,657,678,699,712,733,746,767,788,801,822,843,856,877,890,911,932,945
%N A134864 Wythoff BBB numbers.
%C A134864 The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BBB = 3A+5B.
%C A134864 The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - _Amiram Eldar_, Mar 24 2025
%H A134864 Vincenzo Librandi, <a href="/A134864/b134864.txt">Table of n, a(n) for n = 1..5000</a>
%H A134864 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 A134864 a(n) = B(B(B(n))), n>=1, with B=A001950, the upper Wythoff sequence.
%p A134864 a:=n->floor(n*((1+sqrt(5))/2)^2): [a(a(a(n)))$n=1..55]; # _Muniru A Asiru_, Nov 24 2018
%t A134864 Nest[Quotient[#(3+Sqrt@5),2]&,#,3]&/@Range@100 (* _Federico Provvedi_, Nov 24 2018 *)
%t A134864 b[n_]:=Floor[n GoldenRatio^2]; a[n_]:=b[b[b[n]]]; Array[a, 60] (* _Vincenzo Librandi_, Nov 24 2018 *)
%o A134864 (Python)
%o A134864 from sympy import floor
%o A134864 from mpmath import phi
%o A134864 def B(n): return floor(n*phi**2)
%o A134864 def a(n): return B(B(B(n))) # _Indranil Ghosh_, Jun 10 2017
%o A134864 (Python)
%o A134864 from math import isqrt
%o A134864 def A134864(n): return (m:=5*n)+(((n+isqrt(n*m))&-2)<<2) # _Chai Wah Wu_, Aug 10 2022
%Y A134864 Cf. A000201, A001950, A003622, A003623, A035336, A094214, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134863, A035513.
%Y A134864 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 A134864 nonn
%O A134864 1,1
%A A134864 _Clark Kimberling_, Nov 14 2007