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A076092
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a(n) = n - 2*Sum_{i=1..n} b(i) (see comment for definition of b(i)).
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2
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1, 0, -1, 0, 1, 2, 1, 0, -1, -2, -1, -2, -3, -2, -1, 0, -1, 0, 1, 2, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 0, -1, -2, -3, -4, -5, -6, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 1, 0, -1, -2, -3, -4, -5, -4, -5, -4, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -4, -5, -4, -3
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OFFSET
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1,6
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COMMENTS
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Recall the modified Collatz map: x->x/2 if x is even; x->(3x+1)/2 if x is odd. Let C_m(n) denotes the image of n after m iterations. Then b(n) = (lim_{k->infinity} C_2k(n))-1 (from the Collatz conjecture C_2k(n) is constant = 1 or 2 for k sufficiently large).
Curiously the graph of a(n) has "regularities" around 0 and a pattern that becomes larger and larger when compared with a random sequence of the form n - 2*Sum_{k=1..n} r(k) where r(k) takes random values from (0;1).
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LINKS
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EXAMPLE
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b(12)=1 since, starting with 12, the Collatz map gives: 12->6->3->5->8->4->2->1, then C_6(12)=2 and then b(12) = C_6(12)-1 = 1.
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PROG
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(PARI) a(n)=n-2*sum(i=1, n, if(i<0, 0, s=i; c=0; while(s>1, s=(s%2)*(3*s+1)/2+(1-s%2)*s/2; c++); c)%2)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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