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%I #15 Jan 15 2017 09:16:11
%S 3,5,8,13,45,82,127,209,717,1307,2024,3331,11427,20830,32257,53087,
%T 182115,331973,514088,846061,2902413,5290738,8193151,13483889,
%U 46256493,84319835,130576328,214896163,737201475,1343826622,2081028097,3424854719,11748967107
%N Numerators 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="/A259597/b259597.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^7-x^6+2*x^5-3*x^4+13*x^3+8*x^2+5*x+3) / (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 v = Numerator[t]
%t LinearRecurrence[{0,0,0,16,0,0,0,-1},{3,5,8,13,45,82,127,209},40] (* _Harvey P. Dale_, Jan 15 2017 *)
%o (PARI) Vec((x^7-x^6+2*x^5-3*x^4+13*x^3+8*x^2+5*x+3)/(x^8-16*x^4+1) + O(x^50)) \\ _Colin Barker_, Jul 21 2015
%Y Cf. A041008, A041009, A259596 (denominators).
%K nonn,easy,frac
%O 0,1
%A _Clark Kimberling_, Jul 20 2015
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