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A259596 Denominators of the other-side convergents to sqrt(7).

%I A259596 #17 Jun 03 2017 13:25:55
%S A259596 1,2,3,5,17,31,48,79,271,494,765,1259,4319,7873,12192,20065,68833,
%T A259596 125474,194307,319781,1097009,1999711,3096720,5096431,17483311,
%U A259596 31869902,49353213,81223115,278635967,507918721,786554688,1294473409,4440692161,8094829634
%N A259596 Denominators of the other-side convergents to sqrt(7).
%C A259596 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 A259596 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
%C A259596 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. The 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.
%H A259596 Colin Barker, <a href="/A259596/b259596.txt">Table of n, a(n) for n = 0..1000</a>
%H A259596 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,16,0,0,0,-1).
%F A259596 p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
%F A259596 a(n) = 16*a(n-4) - a(n-8) for n>7. - _Colin Barker_, Jul 21 2015
%F A259596 G.f.: -(x+1)*(x^2-x-1)*(x^4+3*x^2+1) / (x^8-16*x^4+1). - _Colin Barker_, Jul 21 2015
%e A259596 For r = sqrt(7), 3, 5/2, 8/3, 13/5, 45/17, 82/31, 127/48. A comparison of convergents with other-side convergents:
%e A259596 i    p(i)/q(i)         P(i)/Q(i)    p(i)*Q(i)-P(i)*q(i)
%e A259596 0    2/1   < sqrt(7) <    3/1               -1
%e A259596 1    3/1   > sqrt(7) >    5/2                1
%e A259596 2    5/2   < sqrt(7) <    8/3               -1
%e A259596 3    8/3   > sqrt(7) >   13/5                1
%e A259596 4    37/14 < sqrt(7) <   45/17              -1
%e A259596 5    45/17 > sqrt(7) >   83/31               1
%t A259596 r = Sqrt[7]; a[i_] := Take[ContinuedFraction[r, 35], i];
%t A259596 b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
%t A259596 t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
%t A259596 u = Denominator[t]
%t A259596 LinearRecurrence[{0,0,0,16,0,0,0,-1},{1,2,3,5,17,31,48,79},40] (* _Harvey P. Dale_, Jun 03 2017 *)
%o A259596 (PARI) Vec(-(x+1)*(x^2-x-1)*(x^4+3*x^2+1)/(x^8-16*x^4+1) + O(x^50)) \\ _Colin Barker_, Jul 21 2015
%Y A259596 Cf. A041008, A041009, A259597 (numerators).
%K A259596 nonn,easy,frac
%O A259596 0,2
%A A259596 _Clark Kimberling_, Jul 20 2015