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%I #23 Dec 22 2023 12:11:03
%S 1,3,11,29,109,287,1079,2841,10681,28123,105731,278389,1046629,
%T 2755767,10360559,27279281,102558961,270037043,1015229051,2673091149,
%U 10049731549,26460874447,99482086439,261935653321,984771132841,2592895658763,9748229241971
%N Denominators 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: 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="/A259594/b259594.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) = Q(i).
%F a(n) = 10*a(n-2) - a(n-4) for n>3. - _Colin Barker_, Jul 21 2015
%F G.f.: -(x+1)*(x^2-2*x-1) / (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*)
%o (PARI) Vec(-(x+1)*(x^2-2*x-1)/(x^4-10*x^2+1) + O(x^50)) \\ _Colin Barker_, Jul 21 2015
%Y Cf. A041006, A041007, A259595.
%K nonn,easy,frac
%O 0,2
%A _Clark Kimberling_, Jul 20 2015
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