By Shalosh B. Ekhad
[This computer-generated article is an appendix to Doron Zeilberger's talk, delivered on Sept. 15, 2016, at the Rutgers University Experimental Mathematics Seminar, and may be viewed here ]
Each iteration of the The Salamin-Brent algorithm for computing Pi, starting at k=1, gives the following number of (decimal) digits
[1, 4, 9, 20, 42, 85, 173, 347, 697, 1395, 2792, 5587, 11175, 22352, 44706, 89414, 178830]
Note that the first 9 entries are listed (page 5) in the article
by David B. Bailey, Jonathan M. Borwein, Peter B. Borwein, and Simon M. Ploufe, that appeared in the print-magazine "Mathematical Intelligencer", vol. 19, no. 1 (Jan. 1997), pg. 50-57.
The analogous sequences for the Borwein brothers' cubic and quartic algorithms for 1/Pi, mentioned in the above-mentioned article seem to be new.
The Borwein brothers' amazing algorithms are described at length in their classic book, "Pi and the AGM: A study in Analytic Number Theory and the Computational Complexity" , John Wiley, 1987, where references to the original articles can be found.
Each iteration of the The Cubic Borwein-Borwein algorithm for computing 1/Pi, starting at k=1, gives the following number of (decimal) digits
[6, 22, 71, 218, 659, 1985, 5963, 17898, 53704, 161124]
Each iteration of the The Quartic Borwein-Borwein algorithm for computing 1/Pi, starting at k=1, gives the following number of (decimal) digits
[9, 41, 171, 694, 2790, 11172, 44702, 178825]
This ends this article, that took, 10472.122, seconds. to generate.