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A316911 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)*((1-t)^2*t^2/(1+t))^n*dt and write K(n) = d(n)*log(2) - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).

%I A316911 #16 Aug 21 2018 22:52:55
%S A316911 0,25,1719,143731,64456699,1846991851,781688106621,445837607665267,
%T A316911 611642484654021,674842075634295726569,9142845536119405749427,
%U A316911 38984536004906714808649,80321414381403813427242343,342487507476162248453574514441,562411667990487545372378396727201
%N A316911 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)*((1-t)^2*t^2/(1+t))^n*dt and write K(n) = d(n)*log(2) - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).
%C A316911 As n goes to infinity, integral value K(n) goes to zero. Given a rational approximant r(n)=a(n)/c(n)/d(n)=p(n)/q(n) to irrational number log(2), the quality M(n) is defined as, M(n)=-log(|r(n)-log(2)|)/log(q(n)) (Cf. Beukers Link). For this approximation, we can easily measure M(n) over n=5,000..20,000, and estimate that M(n)~1.14... to the 99% confidence level (Cf. Histogram Link).
%H A316911 F. Beukers, <a href="https://dspace.library.uu.nl/handle/1874/26398">A rational approach to Pi</a>, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 378.
%H A316911 Bradley Klee, <a href="/A316911/a316911.png">Quality Histogram</a>.
%F A316911 Define G(x) = Sum_{n>0} A316911(n)/A316912(n)*x^n, and G^(k)(x) = d^k/dx^k G(x). Period G(x) satisfies a nonhomogeneous differential equation: -225+112*x = Sum_{j=0..5,k=0..3} M_{j,k} x^j G^(k)(x), with integer matrix M as in A190726.
%e A316911 {a(10),c(10),d(10)}={9142845536119405749427,307660953600,42872967012}.
%e A316911 r(10)=a(10)/c(10)/d(10)=9142845536119405749427/13190337914573262643200.
%e A316911 r(10)=0.693147180559945309417232121402...
%e A316911 log(2)=0.693147180559945309417232121458...
%e A316911 M(10)=-log(|r(10)-log(2)|)/log(13190337914573262643200)=1.27...
%t A316911 FracData[n0_]:=RecurrenceTable[{2*(n-1)*(2*n-3)*(2*n-1)*(33*n-8)*a[n-2]+ 9*(2*n-1)*(693*n^3-1554*n^2+989*n-160)*a[n-1] -3*n*(3*n-2)*(3*n-1)*(33*n-41)*a[n] == 0, a[0]==0, a[1]==25/6}, a, {n, 0, n0}]
%t A316911 Numerator[FracData[5000]]
%Y A316911 Integer Part: A190726. Denominators: A316912. Similar Pi approximation: A123178, A305997, A305998.
%K A316911 nonn,frac
%O A316911 0,2
%A A316911 _Bradley Klee_, Jul 16 2018