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Wythoff BBB numbers.
11

%I #28 Aug 11 2022 03:20:28

%S 13,34,47,68,89,102,123,136,157,178,191,212,233,246,267,280,301,322,

%T 335,356,369,390,411,424,445,466,479,500,513,534,555,568,589,610,623,

%U 644,657,678,699,712,733,746,767,788,801,822,843,856,877,890,911,932,945

%N Wythoff BBB numbers.

%C The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BBB=3A+5B.

%H Vincenzo Librandi, <a href="/A134864/b134864.txt">Table of n, a(n) for n = 1..5000</a>

%H 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 a(n) = B(B(B(n))), n>=1, with B=A001950, the upper Wythoff sequence.

%p a:=n->floor(n*((1+sqrt(5))/2)^2): [a(a(a(n)))$n=1..55]; # _Muniru A Asiru_, Nov 24 2018

%t Nest[Quotient[#(3+Sqrt@5),2]&,#,3]&/@Range@100 (* _Federico Provvedi_, Nov 24 2018 *)

%t b[n_]:=Floor[n GoldenRatio^2]; a[n_]:=b[b[b[n]]]; Array[a, 60] (* _Vincenzo Librandi_, Nov 24 2018 *)

%o (Python)

%o from sympy import floor

%o from mpmath import phi

%o def B(n): return floor(n*phi**2)

%o def a(n): return B(B(B(n))) # _Indranil Ghosh_, Jun 10 2017

%o (Python)

%o from math import isqrt

%o def A134864(n): return (m:=5*n)+(((n+isqrt(n*m))&-2)<<2) # _Chai Wah Wu_, Aug 10 2022

%Y Cf. A000201, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134863, A035513.

%Y 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 nonn

%O 1,1

%A _Clark Kimberling_, Nov 14 2007