This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A259593 #23 Aug 20 2021 18:21:40
%S A259593 2,3,7,12,26,45,97,168,362,627,1351,2340,5042,8733,18817,32592,70226,
%T A259593 121635,262087,453948,978122,1694157,3650401,6322680,13623482,
%U A259593 23596563,50843527,88063572,189750626,328657725,708158977,1226567328,2642885282,4577611587
%N A259593 Numerators of the other-side convergents to sqrt(3).
%C A259593 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 A259593 p(i)/q(i) = [a(0), a(1), ..., a(i)] and
%C A259593 P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1].
%C A259593 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.
%C A259593 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 A259593 Colin Barker, <a href="/A259593/b259593.txt">Table of n, a(n) for n = 0..1000</a>
%H A259593 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-1).
%F A259593 p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
%F A259593 a(n) = 4*a(n-2) - a(n-4) for n>3. - _Colin Barker_, Jul 21 2015
%F A259593 G.f.: -(x^2-3*x-2) / (x^4-4*x^2+1). - _Colin Barker_, Jul 21 2015
%F A259593 a(n) = 3^(n/2 - t + 1)*((2 + sqrt(3))^t + (-1)^n*(2 - sqrt(3))^t)/2, where t = floor(n/2) + 1. - _Ridouane Oudra_, Aug 03 2021
%e A259593 For r = sqrt(3), the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532. A comparison of convergents with other-side convergents:
%e A259593 i p(i)/q(i) P(i)/Q(i) p(i)*Q(i) - P(i)*q(i)
%e A259593 0 1/1 < sqrt(3) < 2/1 -1
%e A259593 1 2/1 > sqrt(3) > 3/2 1
%e A259593 2 5/3 < sqrt(3) < 7/4 -1
%e A259593 3 7/4 > sqrt(3) > 12/7 1
%e A259593 4 19/11 < sqrt(3) < 26/15 -1
%e A259593 5 26/15 > sqrt(3) > 45/26 1
%t A259593 r = Sqrt[3]; a[i_] := Take[ContinuedFraction[r, 35], i];
%t A259593 b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
%t A259593 t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
%t A259593 v = Numerator[t]
%o A259593 (PARI) Vec(-(x^2-3*x-2)/(x^4-4*x^2+1) + O(x^50)) \\ _Colin Barker_, Jul 21 2015
%Y A259593 Cf. A002530, A002531, A259592 (denominators).
%Y A259593 Cf. A001075 (even bisection), A005320 (odd bisection).
%K A259593 nonn,easy,frac
%O A259593 0,1
%A A259593 _Clark Kimberling_, Jul 20 2015