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%I #10 Feb 24 2023 15:54:55
%S 1,8,113,219,33215,66317,99532,165849,364913,630294,1725033,3085153,
%T 27235615,78256779,131002976,209259755,471265707,1151791169,
%U 2774848045,6701487259,11439654911,574364584667,1709690779483,2851718461558,4561409241041,47337186164411
%N Denominators of the other-side convergents to Pi.
%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.
%C 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.
%e For r = Pi, the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532.
%e 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 3/1 < Pi < 4/1 -1
%e 1 22/7 > Pi > 25/8 1
%e 2 333/106 < Pi < 355/113 -1
%t r=Pi;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 Denominator[t] (* A259590 *)
%t Numerator[t] (* A259591 *)
%Y Cf. A259591, A002485, A002486, A259588.
%K nonn,easy,frac
%O 0,2
%A _Clark Kimberling_, Jul 17 2015
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