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A283188
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A periodic sequence of 8-bit binary numbers for single-bit multi-frequency generation.
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1
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255, 127, 191, 31, 207, 71, 163, 33, 240, 80, 152, 24, 236, 108, 174, 6, 215, 87, 179, 51, 235, 73, 137, 9, 252, 116, 180, 20, 198, 70, 170, 42, 251, 91, 155, 17, 229, 101, 165, 5, 220, 92, 186, 58, 234, 66, 130, 2, 247, 117, 189, 29, 205, 77, 169, 33, 242, 82, 146, 18, 238, 110, 174, 12, 221
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OFFSET
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0,1
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COMMENTS
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If the updating frequency of numbers is f, then the most significant bit (MSB) changes with frequency f, while the second MSB changes with f/2, the third MSB changes with f/3, fourth MSB with f/4, and so on till the least significant bit (LSB), which changes with f/8.
The length of the cyclic period is 1680 = lcm(2,4,6,8,10,12,14,16). Only 192 numbers are used out of the 256 possible 8-bit binary numbers. Possible number of repetitions within a cycle are 6, 8, 9, and 12.
For a fast continuous production of the sequence, it could be more efficient to implement a cyclic lookup table rather than calculating the numbers each time.
A similar sequence can be obtained from A272614 (excluding the first 8 terms) by selecting the 8 most significant bits after having removed the MSB.
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LINKS
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FORMULA
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a(n) = Sum_{k = 1..8} floor((n - k)/k) mod 2))*2^(8 - k).
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EXAMPLE
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Binary expansions of the first 16 terms:
[1, 1, 1, 1, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1, 1, 1]
[1, 0, 1, 1, 1, 1, 1, 1]
[0, 0, 0, 1, 1, 1, 1, 1]
[1, 1, 0, 0, 1, 1, 1, 1]
[0, 1, 0, 0, 0, 1, 1, 1]
[1, 0, 1, 0, 0, 0, 1, 1]
[0, 0, 1, 0, 0, 0, 0, 1]
[1, 1, 1, 1, 0, 0, 0, 0]
[0, 1, 0, 1, 0, 0, 0, 0]
[1, 0, 0, 1, 1, 0, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0]
[1, 1, 1, 0, 1, 1, 0, 0]
[0, 1, 1, 0, 1, 1, 0, 0]
[1, 0, 1, 0, 1, 1, 1, 0]
[0, 0, 0, 0, 0, 1, 1, 0]
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MATHEMATICA
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a[n_] := Sum[Floor@Mod[(n - k)/k, 2]*2^(8 - k), {k, 1, 8}];
Table[a[n], {n, 0, 64}]
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PROG
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(PARI) a(n) = sum(k=1, 8, ((floor((n - k) / k)) % 2)*2^(8 - k)); \\ Indranil Ghosh, Mar 03 2017
(Python)
....s=0
....for k in range(1, 9):
........s+=(((n-k)/k)%2) * 2**(8-k)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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