A259590 - OEIS

%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