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A331101
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Denominators of the best approximations for sqrt(2).
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2
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1, 2, 3, 5, 12, 17, 29, 70, 99, 169, 408, 577, 985, 2378, 3363, 5741, 13860, 19601, 33461, 80782, 114243, 195025, 470832, 665857, 1136689, 2744210, 3880899, 6625109, 15994428, 22619537, 38613965, 93222358, 131836323, 225058681, 543339720, 768398401, 1311738121, 3166815962
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OFFSET
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1,2
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COMMENTS
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Let w = sqrt(2). Each of the principal convergents 1/1, 3/2, 7/5, 17/12, ..., see A002965, represents a best approximation for w because no other fraction with a smaller denominator is closer to w.
However, with abs(3/2 - w) > abs(4/3 - w)> abs(7/5 - w), 4/3 is another best approximation which has to be inserted. Generally, after each principal convergent p/q, we must insert the correspondent intermediate convergent 2q/p = (2/1), 4/3, (10/7), 24/17, ..., if it is closer to w than p/q (terms without brackets).
It is a well-known fact that the geometric mean sqrt(a*b) of two factors a and b is closer to the smaller one. As w is the geometric mean of p/q and 2q/p, the second term is inserted if it is smaller than p/q. This applies to every second intermediate convergent because the principal convergents alternately undershoot and overshoot w.
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LINKS
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FORMULA
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If n mod 3 = 2: a(n) = 3*a(n - 1) - a(n - 2), otherwise: a(n) = a(n - 1) + a(n - 2), with a(1) = 1, a(2) = 2.
a(3n - 2) = w/4*D(2n - 1), a(3n - 1) = w/4*D(2n), a(3n) = 1/2*S(2n), for n>0 with w = sqrt(2) and S(n) = (1 + w)^n + (1 - w)^n and D(n) = (1 + w)^n - (1 - w)^n.
G.f.: x*(1 + 2*x + 3*x^2 - x^3 - x^5) / (1 - 6*x^3 + x^6).
a(n) = 6*a(n - 3) - a(n - 6) for n > 6.
(End)
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EXAMPLE
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The fractions are 1/1, 3/2, 4/3, 7/5, 17/12, 24/17, ...
Let w = sqrt(2) again. The first four principal convergents are, see comments, 1/1 (which is less than w), 3/2 (greater than w), 7/5 (less than w), 17/12 (greater than w). After 3/2, the fraction 2 * 2/3 = 4/3 is inserted because 4/3 < 3/2 and therefore w - 4/3 < 3/2 - w (0.081... < 0.085...). After 7/5, the fraction 2 * 5/7 = 10/7 is not inserted, because 10/7 > 7/5 etc.
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PROG
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(PARI) Vec(x*(1 + 2*x + 3*x^2 - x^3 - x^5) / (1 - 6*x^3 + x^6) + O(x^40)) \\ Colin Barker, Jan 09 2020
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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