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%I #27 Apr 14 2018 08:19:16
%S 3,7,113,4739,46804,134370,614063,1669512,15474115,18858140,19180902,
%T 41486462,492988666,1794101482,34644610027,48670872793,97414216753,
%U 138669015304,195575194804,543142431219,3173502039447,4968328076747
%N A modified Pierce-type expansion for Pi: Pi = a(0) + Sum_{n>=1} (-1)^floor(n/2)/(Product_{i=1..n} a(i)).
%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%F a(0) = floor(Pi); A(1) = Pi-a(0); a(2*n-1) = floor(1/A(2*n-1)); A(2*n) = 1-a(2*n-1)*A(2*n-1); a(2*n) = ceiling(1/A(2*n)) and A(2*n+1) = a(2*n)*A(2*n)-1 for n >= 1.
%e Pi = 3 + 1/7 - 1/(7*113) - 1/(7*113*4739) + 1/(7*113*4739*46804) + 1/(7*113*4739*46804*134370) - 1/(7*113*4739*46804*134370*614063) - 1/(7*113*4739*46804*134370*614063*1669512) + ...
%e From _M. F. Hasler_, Apr 09 2018: (Start)
%e Using the formulas given in the formula section, we get:
%e a(0) = 3, A(1) = Pi-3 = 0.14159..., a(1) = floor(1/A(1)) = floor(7.0626...) = 7,
%e A(2) = 1 - A(1)*a(1) = 0.00885..., a(2) = ceiling(1/A(2)) = 113,
%e A(3) = A(2)*a(2) - 1 = 0.000221..., a(3) = floor(1/A(3)) = 4739,
%e A(4) = 1 - A(3)*a(3) = 2.136585...e-5, a(4) = ceiling(1/A(2)) = 46804,
%e A(5) = A(4)*a(4) - 1 = 7.442125...e-7, a(5) = floor(1/A(3)) = 134370, ... (End)
%o (PARI) {A=Pi-a=3; for(n=0,oo, print1(a","); A=abs(1-A*a=if(bittest(n,0),ceil(1/A),1\A)))} \\ _M. F. Hasler_, Apr 09 2018
%Y Cf. A061233.
%K nonn
%O 0,1
%A Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 02 2000
%E Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 28 2001
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