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A272170
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Second most significant bit of Fibonacci numbers > 1 written in base 2.
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8
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0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0
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OFFSET
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3,1
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COMMENTS
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It is conjectured that there are no more than two consecutive "0's" or “1’s” (tested up to n=10^5). The sequence looks quasiperiodic and its Fourier spectrum seems to have a fractal structure.
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LINKS
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FORMULA
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a(n) = floor(A000045(n)/(2^(ceiling(log_2(A000045(n) + 1)) - 2))) - 2.
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EXAMPLE
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(second MSB in parenthesis)
3 2 -> 1(0)
4 3 -> 1(1)
5 5 -> 1(0)1
6 8 -> 1(0)00
7 13 -> 1(1)01
8 21 -> 1(0)101
9 34 -> 1(0)0010
10 55 -> 1(1)0111
...
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MATHEMATICA
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nmax = 120; Table[IntegerDigits[Fibonacci[j], 2][[2]], {j, 3, nmax}]
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PROG
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(PARI) a(n) = binary(fibonacci(n))[2]; \\ Michel Marcus, Apr 25 2016
(Python)
for n in range(3, 10001):
a, b = b, a+b
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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