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A259591 Numerators of the other-side convergents to Pi.

%I A259591 #6 Jul 19 2015 11:04:46
%S A259591 4,25,355,688,104348,208341,312689,521030,1146408,1980127,5419351,
%T A259591 9692294,85563208,245850922,411557987,657408909,1480524883,3618458675,
%U A259591 8717442233,21053343141,35938735828,1804419559672,5371151992734,8958937768937,14330089761671
%N A259591 Numerators of the other-side convergents to Pi.
%C A259591 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 A259591 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 A259591 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 A259591 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.
%F A259591 p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
%e A259591 For r = Pi, the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532.
%e A259591 A comparison of convergents with other-side convergents:
%e A259591 i  p(i)/q(i)        P(i)/Q(i)  p(i)*Q(i) - P(i)*q(i)
%e A259591 0     3/1    < Pi <    4/1               -1
%e A259591 1    22/7    > Pi >    25/8               1
%e A259591 2   333/106  < Pi <    355/113           -1
%t A259591 r = Pi; a[i_] := Take[ContinuedFraction[r, 35], i];
%t A259591 b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
%t A259591 t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
%t A259591 u = Denominator[t]  (*A259590*)
%t A259591 v = Numerator[t]    (*A259591*)
%Y A259591 Cf. A259590, A002485, A002486.
%K A259591 nonn,easy,frac
%O A259591 0,1
%A A259591 _Clark Kimberling_, Jul 17 2015