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A137323
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Characteristic value of numbers used to compute number of binary expansions of a certain length that have a given number of rotational symmetries.
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0
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0, 0, 0, 1, 0, 4, 0, 8, 3, 17, 0, 42, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Sequence is useful in counting binary expansions of length l with r rotational symmetries (we only consider r>1), where r has to be a proper divisor not equal to 1.
We discount numbers with l symmetries, because we know this only occurs once, at (2^l) - 1.
For example, consider binary expansions of length 8. We know that for any number the possible symmetries are the proper divisors of 8 not equal to 1; (2, 4).
So if we would like to find the number of expansions of length 8 that have 2 rotational symmetries, it is [2^(8/2 -1) -1] - a(8/2) = 7 - 1 = 6.
In general it appears that the formula for r rotational-symmetric numbers of expansion length l is the following: [2^(l/r -1) -1] - a(l/r).
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REFERENCES
| Fraleigh, John B. "A first course in abstract algebra". Pearson Education, 2003.
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FORMULA
| a(x) = Sum|(s is a proper divisor of x not equal to one)| 2^((x/s -1)) -1.
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EXAMPLE
| a(6) = 4 because 6 has (2,3) as proper divisors neq to one. Plugging these values into the formula we get [2(6/2 -1)-1] + [2(6/3 -1) -1] = 3 + 1 = 4.
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CROSSREFS
| Cf. A138904.
Sequence in context: A010638 A123961 A020763 * A021075 A109169 A011291
Adjacent sequences: A137320 A137321 A137322 * A137324 A137325 A137326
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KEYWORD
| base,nonn,uned
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AUTHOR
| Max Sills (maxwell.sills(AT)case.edu), Apr 06 2008
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