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A134860 Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.

%I A134860 #42 Mar 29 2025 12:14:36
%S A134860 4,12,17,25,33,38,46,51,59,67,72,80,88,93,101,106,114,122,127,135,140,
%T A134860 148,156,161,169,177,182,190,195,203,211,216,224,232,237,245,250,258,
%U A134860 266,271,279,284,292,300,305,313,321,326,334,339,347,355,360,368,373
%N A134860 Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.
%C A134860 The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAB=AA+AB and AAB=A+2B-1.
%C A134860 The asymptotic density of this sequence is 1/phi^4 = 2/(7+3*sqrt(5)), where phi is the golden ratio (A001622). - _Amiram Eldar_, Mar 21 2022
%H A134860 Amiram Eldar, <a href="/A134860/b134860.txt">Table of n, a(n) for n = 1..10000</a>
%H A134860 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 A134860 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., Vol. 24, No. 2 (2010), pp. 570-588. - _N. J. A. Sloane_, May 06 2011
%H A134860 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html">Complementary equations and Wythoff Sequences</a>, Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.3.
%F A134860 a(n) = A(A(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.
%t A134860 With[{r = Map[Fibonacci, Range[2, 14]]}, Position[#, {1, 0, 1}][[All, 1]] &@ Table[If[Length@ # < 3, {}, Take[#, -3]] &@ IntegerDigits@ Total@ Map[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] &,Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 373}]] (* _Michael De Vlieger_, Jun 09 2017 *)
%o A134860 (Python)
%o A134860 from sympy import fibonacci
%o A134860 def a(n):
%o A134860     x=0
%o A134860     while n>0:
%o A134860         k=0
%o A134860         while fibonacci(k)<=n: k+=1
%o A134860         x+=10**(k - 3)
%o A134860         n-=fibonacci(k - 1)
%o A134860     return x
%o A134860 def ok(n): return str(a(n))[-3:]=="101"
%o A134860 print([n for n in range(4, 501) if ok(n)]) # _Indranil Ghosh_, Jun 08 2017
%o A134860 (Python)
%o A134860 from math import isqrt
%o A134860 def A134860(n): return 3*(n+isqrt(5*n**2)>>1)+(n<<1)-1 # _Chai Wah Wu_, Aug 10 2022
%Y A134860 Cf. A000201, A001622, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
%Y A134860 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 A134860 Set-wise difference A003622 \ A095098. Cf. A095089 (fib101 primes).
%K A134860 nonn,base
%O A134860 1,1
%A A134860 _Antti Karttunen_, Jun 01 2004 and _Clark Kimberling_, Nov 14 2007
%E A134860 This is the result of merging two sequences which were really the same. - _N. J. A. Sloane_, Jun 10 2017