Hi, S. K. Lukas in
http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf> derived several identities, which relate Pi (via definite integrals) with the several few Pi fractional convergents, which denominators and numerators are described in http://www.research.att.com/~njas/sequences/A002486<http://www.research.att.com/%7Enjas/sequences/A0024865> http://www.research.att.com/~njas/sequences/A002485<http://www.research.att.com/%7Enjas/sequences/A0024865> I raised the issue re 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 (referenced in above sequences) - see 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<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf> Quoting William Stein's kind reply to me: "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." May one of you could help me by modifying (porting to SAGE) and running Stephen's program towards experimental attempt of deriving desired generalized identity? Thanks, Best Regards, Alexander R. Povolotsky > ------------------------------------------------- > On Tue, Feb 10, 2009 at 8:03 PM, Stephen Lucas <[email protected]> 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 <[email protected]> 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<http://www.math.jmu.edu/%7Elucassk/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/ <http://www.math.jmu.edu/%7Elucassk/> >> Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
