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Numerators of the other-side convergents to Pi.
2

%I #6 Jul 19 2015 11:04:46

%S 4,25,355,688,104348,208341,312689,521030,1146408,1980127,5419351,

%T 9692294,85563208,245850922,411557987,657408909,1480524883,3618458675,

%U 8717442233,21053343141,35938735828,1804419559672,5371151992734,8958937768937,14330089761671

%N Numerators 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.

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

%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 u = Denominator[t] (*A259590*)

%t v = Numerator[t] (*A259591*)

%Y Cf. A259590, A002485, A002486.

%K nonn,easy,frac

%O 0,1

%A _Clark Kimberling_, Jul 17 2015