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A355681
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The "coarser" of 2 representations of the Cantor middle thirds set viewed from a quarter point that lies at a(0) (the third 1 in the data).
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2
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-1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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-13
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COMMENTS
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Each occurrence of the consecutive values -1, 1 represents the 2 quarter points of a scaled image of the Cantor middle thirds set whose endpoints can be considered to lie halfway between these terms and the 0's either side. This occurs in both halves of the figure in the examples, which also illustrates the relevance of the values of the terms.
a(n) is the restriction of a function, f, which is defined for all integers as specified in the formula section. For reasons of presentation, the offset is chosen to be -13. (Beyond -13, the next nonzero value of f is f(-68) = 1.) Essentially though, we can consider f to be giving a microscopic view of the middle thirds set around either quarter point.
f:Z -> {-1,0,1} is used to generate the closure under multiplication by -3 of the scaled Cantor middle thirds set spanning [-3/2, 1/2]. So this process also generates similar Cantor sets spanning [-3/2, 9/2], [-27/2, 9/2], [-27/2, 81/2], ... . 0 is clearly a quarter point in all these intervals.
Formally, define c: Z -> P(R) so that c(n) is the empty set if f(n) = 0, otherwise the translated Cantor middle thirds set centered at n+f(n)/6 and scaled by 2/3. Let C_oo be the union of c(m) for all n in Z. C_oo is the closure under multiplication by -3 of the scaled translated Cantor middle thirds set spanning [-3/2, 1/2].
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LINKS
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Eric Weisstein's World of Mathematics, Cantor Set.
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FORMULA
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a(n) = f(n), where f:Z -> {-1,0,1}; f(0) = 1, for integer m, -1 <= i <= 1, writing x = -3*f(-m) - 6i, f(3m+i) = if |x-4| < 2 then x-4 otherwise if |x+4| < 2 then x+4 otherwise 0.
The generation of f can be understood, in the manner of a morphism, using the following table:
f(-m) -> f(3m-1) f(3m) f(3m+1)
-1 -> 0 -1 1
0 -> 0 0 0
1 -> -1 1 0
a(9n) = a(n).
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EXAMPLE
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Illustration of the generation of the scaled Cantor middle thirds set spanning [-3/2, 9/2], the line titled "Cantor" showing the similar sets of span 2/3 centered at n+a(n)/6 that are mentioned in the comments:
n: -2 -1 0 1 2 3 4 5
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Sixths: :..:..:..:..:..:..:..:..:..:..:..:..:..:..:
"Cantor": |<->| |<->| |<->| |<->|
^ ^ ^ ^
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|< >| |< >|
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a(n): 0 -1 1 0 0 -1 1 0
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Notice that n=0 marks the 1/4 point of the scaled middle thirds set spanning [-1/6, 1/2], the 3/4 point of the similar set spanning [-3/2, 1/2] and the 1/4 point of the set spanning [-3/2, 9/2]. This continues at larger scales, 0 being at the 3/4 point of the similar set spanning [-27/2, 9/2], the 1/4 point of [-27/2, 81/2] and so on. Likewise at smaller scales, n=0 marks the 3/4 point of the similar set spanning [-1/6, 1/18], the 1/4 point of [-1/54, 1/18] etc.
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PROG
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(PARI) a(n) = {if (n==0, 1,
my(x); x = -3*a(-n\/3) - 6*((n+1)%3-1);
if(abs(x-4)<2, x-4, if(abs(x+4)<2, x+4, 0)))}
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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