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%I #17 Mar 21 2023 10:03:38
%S 3,7,27,71,267,703,2643,6959,26163,68887,258987,681911,2563707,
%T 6750223,25378083,66820319,251217123,661452967,2486793147,6547709351,
%U 24616714347,64815640543,243680350323,641608696079,2412186788883,6351271320247,23878187538507
%N Numerators of the other-side convergents to sqrt(6).
%C 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:
%C 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
%C |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.
%H Colin Barker, <a href="/A259595/b259595.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).
%F p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
%F a(n) = 10*a(n-2) - a(n-4) for n>3. - _Colin Barker_, Jul 21 2015
%F G.f.: (x^3-3*x^2+7*x+3) / (x^4-10*x^2+1). - _Colin Barker_, Jul 21 2015
%e 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:
%e i p(i)/q(i) P(i)/Q(i) p(i)*Q(i)-P(i)*q(i)
%e 0 2/1 < sqrt(6) < 3/1 -1
%e 1 5/2 > sqrt(6) > 7/3 1
%e 2 22/9 < sqrt(6) < 27/11 -1
%e 3 49/20 > sqrt(6) > 71/29 1
%e 4 218/89 < sqrt(6) < 267/109 -1
%e 5 485/198 > sqrt(6) > 703/287 1
%t r = Sqrt[6]; a[i_] := Take[ContinuedFraction[r, 35], i];
%t b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
%t t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
%t u = Denominator[t] (* A259594 *)
%t v = Numerator[t] (* A259595 *)
%t LinearRecurrence[{0,10,0,-1},{3,7,27,71},30] (* _Harvey P. Dale_, Mar 21 2023 *)
%o (PARI) Vec((x^3-3*x^2+7*x+3)/(x^4-10*x^2+1) + O(x^50)) \\ _Colin Barker_, Jul 21 2015
%Y Cf. A041006, A041007, A259594.
%K nonn,easy,frac
%O 0,1
%A _Clark Kimberling_, Jul 20 2015
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