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A259595
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Numerators of the other-side convergents to sqrt(6).
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2
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3, 7, 27, 71, 267, 703, 2643, 6959, 26163, 68887, 258987, 681911, 2563707, 6750223, 25378083, 66820319, 251217123, 661452967, 2486793147, 6547709351, 24616714347, 64815640543, 243680350323, 641608696079, 2412186788883, 6351271320247, 23878187538507
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OFFSET
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0,1
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COMMENTS
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Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0. Closeness of P(i)/Q(i) to r is indicated by
|r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.
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LINKS
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FORMULA
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p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
a(n) = 10*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: (x^3-3*x^2+7*x+3) / (x^4-10*x^2+1). - Colin Barker, Jul 21 2015
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EXAMPLE
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For r = sqrt(6), the first 7 other-side convergents are 3, 7/3, 27/11, 71/29, 267/109, 703/287, 2643/1079. A comparison of convergents with other-side convergents:
i p(i)/q(i) P(i)/Q(i) p(i)*Q(i)-P(i)*q(i)
0 2/1 < sqrt(6) < 3/1 -1
1 5/2 > sqrt(6) > 7/3 1
2 22/9 < sqrt(6) < 27/11 -1
3 49/20 > sqrt(6) > 71/29 1
4 218/89 < sqrt(6) < 267/109 -1
5 485/198 > sqrt(6) > 703/287 1
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MATHEMATICA
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r = Sqrt[6]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
LinearRecurrence[{0, 10, 0, -1}, {3, 7, 27, 71}, 30] (* Harvey P. Dale, Mar 21 2023 *)
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PROG
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(PARI) Vec((x^3-3*x^2+7*x+3)/(x^4-10*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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STATUS
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approved
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