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A259596 Denominators of the other-side convergents to sqrt(7). 2

%I

%S 1,2,3,5,17,31,48,79,271,494,765,1259,4319,7873,12192,20065,68833,

%T 125474,194307,319781,1097009,1999711,3096720,5096431,17483311,

%U 31869902,49353213,81223115,278635967,507918721,786554688,1294473409,4440692161,8094829634

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

%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

%C 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 Colin Barker, <a href="/A259596/b259596.txt">Table of n, a(n) for n = 0..1000</a>

%H <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 p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).

%F a(n) = 16*a(n-4) - a(n-8) for n>7. - _Colin Barker_, Jul 21 2015

%F 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 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 i p(i)/q(i) P(i)/Q(i) p(i)*Q(i)-P(i)*q(i)

%e 0 2/1 < sqrt(7) < 3/1 -1

%e 1 3/1 > sqrt(7) > 5/2 1

%e 2 5/2 < sqrt(7) < 8/3 -1

%e 3 8/3 > sqrt(7) > 13/5 1

%e 4 37/14 < sqrt(7) < 45/17 -1

%e 5 45/17 > sqrt(7) > 83/31 1

%t r = Sqrt[7]; 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]

%t 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 (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 Cf. A041008, A041009, A259597 (numerators).

%K nonn,easy,frac

%O 0,2

%A _Clark Kimberling_, Jul 20 2015

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Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)