I am looking for help/cooperation in experimental researching of the
possibility of deriving generalized ("n" parameter based ) definite
integral identity relating Pi with ALL (each at its own value of n)
Pi fractional convergents described inhttp://www.research.att.com/~njas/sequences/A002486 http://www.research.att.com/~njas/sequences/A002485 for n=2,3,4,... (excluding n=0,1) See below my exchange with S. K. Lukas. Unfortunately I do not have sufficient computational resources (I do not have access to Maple or Mathematica, instead I have Pari/GP installed on my very old home computer) to take advantage of Stephen's generous offer to play with his Maple > program, which he wrote and which is listed in http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf. May one of you could help me by modifying (porting to SAGE) and running Stephen's program towards verifying is there a pattern or not ? Quotting William Stein's kind reply to my email: "the program is on page 9 of the linked to pdf and that it is literally only 18 lines of code, and would likely be very easy for somebody who knows *both* Sage and Maple to port to Sage." Thanks, Best Regards, Alexander R. Povolotsky > ------------------------------------------------- > On Tue, Feb 10, 2009 at 8:03 PM, Stephen Lucas <luca...@jmu.edu> wrote: >> Alexander, >> Thanks for your email, I could only get to it now due to a heavy day >> of teaching. >> >> I am tempted to say that what you have found are essentially >> coincidences. >> >> However, don't let me dissuade you from searching for patterns -- I >> may indeed have missed them. If you want to experiment with my Maple >> code, go right ahead. >> >> Incidentally, since you read the paper from my website, you may be >> interested to know that an edited version of it (slightly condensed >> without the Maple code) was published this month (February 2009) in >> the American Mathematical Monthly. >> >> Yours sincerely, >> Steve Lucas >> >> On Mon, Feb 9, 2009 at 8:00 PM, Alexander Povolotsky <apovo...@gmail.com> >> wrote: >>> Hello Dear Stephen, >>> >>> I found and read your article "Integral approximations to Pi with >>> nonnegative integrands" >>> http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf >>> >>> I looked specifically into the "make-up" of your formula >>> >>> Pi = 355/113-1/3164*Int(x^8*(1-x)^8*(25+816*x^2)/(1+x^2),x = 0 .. 1) >>> >>> I noticed that >>> 3164 / (816 - 25) = 3164/791= 3164 / (7 * 113) = 4 >>> where originally (incorrectly) I thought that 7 is den of 22/7 and >>> 113 is den of 355/113 ... >>> also we observe that 113 - 7 = 106 >>> >>> Then I looked at your another formula >>> >>> Pi = 104348/33215 -1/38544*Int(x^12*(1-x)^12*(1349 >>> -1060*x^2)/(1+x^2),x = 0 .. 1) >>> >>> I noticed that >>> 38544 / (1349 + 1060) = 38544 / 2409 = 38544 / (3 * 11 * 73 ) = 16 >>> is it interesting that 33 + 73 = 106 as above ? >>> >>> Also one could see that in your formula for 103993/33102 >>> >>> 755216 / (124360 - 77159) = 755216 / 47201 = 755216 / (7 * 11 * 613) = 16 >>> >>> as well as that in your formula for 333/106 >>> >>> 530 / (462 - 197) = 530 / 265 = 530 / (5 * 53) = 2 >>> >>> Is it some sort of pattern there with regards to ratios ? >>> >>> Actually the construct to get the ratios is the same - (if i am not >>> mistaken) it takes in account the initial sign of the involved terms. >>> >>> >>> Thanks, >>> Regards, >>> Alexander R. Povolotsky >> -- >> Stephen Lucas, Associate Professor >> Department of Mathematics and Statistics >> MSC 1911, James Madison University, Harrisonburg, VA 22807 USA >> Phone 540 568 5104, Fax 540 568 6857, Web http://www.math.jmu.edu/~lucassk/ >> Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
