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A098457
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Farey Bisection Expansion of sqrt(7).
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2
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1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1
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OFFSET
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1,1
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COMMENTS
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We define the Farey Bisection Expansion (FBE) of the nonnegative real number x to be the sequence {a(n)} of 0's and 1's determined as follows. Set na(0)=0, da(0)=1, nb(0)=1 and db(0)=0. For n=1, 2, 3,..., set num=na(n-1)+nb(n-1) and den=da(n-1)+db(n-1); if x<n/b, set a(n)=0, na(n)=na(n-1), da(n)=da(n-1), nb(n)=num, db(n)=den, else set a(n)=1, na(n)=num, da(n)=den, nb(n)=nb(n-1), db(n)=db(n-1). (The process is akin to that of locating the zero of a function by the bisection method, simply recording which successive subinterval, the left or the right, the zero lies at each refinement.) The FBE of Sqrt[7] is periodic with period 7. The RUNS transform of FBE(x) is the sequence of partial quotients of the continued fraction of x. As can be seen, RUNS(FBE(Sqrt[7]))={2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1,...}, which is A010121.
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LINKS
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FORMULA
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G.f.: x * (1 + x + x^3 + x^5 + x^6) / (1 - x^7).
a(n) = a(n-7) for n>7.
a(n) = 1 - Sum_{k=1..4} floor((n + k)/7)*(-1)^k. (End)
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EXAMPLE
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G.f. = x + x^2 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^13 + x^14 + x^15 + ...
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MAPLE
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MATHEMATICA
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LinearRecurrence[{0, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 0, 1, 1}, 105] (* Ray Chandler, Aug 26 2015 *)
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PROG
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(PARI) {a(n) = [1, 1, 0, 1, 0, 1, 1][(n-1)%7+1]}; /* Michael Somos, Dec 26 2016 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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