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A134860
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Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.
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14
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4, 12, 17, 25, 33, 38, 46, 51, 59, 67, 72, 80, 88, 93, 101, 106, 114, 122, 127, 135, 140, 148, 156, 161, 169, 177, 182, 190, 195, 203, 211, 216, 224, 232, 237, 245, 250, 258, 266, 271, 279, 284, 292, 300, 305, 313, 321, 326, 334, 339, 347, 355, 360, 368, 373
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OFFSET
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1,1
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COMMENTS
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The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAB=AA+AB and AAB=A+2B-1.
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
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LINKS
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FORMULA
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a(n) = A(A(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.
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MATHEMATICA
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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 *)
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PROG
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(Python)
from sympy import fibonacci
def a(n):
x=0
while n>0:
k=0
while fibonacci(k)<=n: k+=1
x+=10**(k - 3)
n-=fibonacci(k - 1)
return x
def ok(n): return str(a(n))[-3:]=="101"
print([n for n in range(4, 501) if ok(n)]) # Indranil Ghosh, Jun 08 2017
(Python)
from math import isqrt
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CROSSREFS
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Cf. A000201, A001622, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
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.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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This is the result of merging two sequences which were really the same. - N. J. A. Sloane, Jun 10 2017
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STATUS
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approved
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