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 My favorite integer sequence jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/11/2014, 05:19 PM (08/11/2014, 04:51 PM)sheldonison Wrote: Nice post. What is the most accurate version of the $\alpha_1$ scaling constant that you know of? Earlier, you posted, "1.083063". I'm not having much luck finding references to the constant online. The closest two I've seen are the 1.083063 that I found earlier through a google search (post #3 in the current discussion), and a reference to 1/a_1 of 0.9233, on this page: https://oeis.org/A002577     The latter has 4 sig-figs, though it's accurate to almost 5 (1/1.083063 ~= 0.923307) I found that 0.9233 by accident, searching the OEIS to see if they had an integer sequence for the 2^m terms of A(n). Interestingly, I see some of my ideas listed on the page for that integer sequence, so I suppose that means I'm on the right track? I'm just a few decades late to the party. The code I posted earlier will generate arbitrary terms in the sequence with indexes of 2^m, in approximately O(m^5) time. I.e., I can generate terms A(1), A(2), A(4), A(8 ), ..., A(2^m) The code can be modified to calculate an arbitrary term, e.g., A(12345678987665321385) or whatever I'd like, with only a modest increase in running time. However, it becomes far less flexible (it's a one-shot deal, all that effort to calculate a single term), so I haven't implemented it yet in code. (I've used it in Excel for modestly-sized terms, so I know it works.) I'm currently working on a version that would allow me to calculate all the terms of the form [(2^(n-1)+1)...(2^n)]*(2^m), for a given small n and arbitrary m, which should run in about O(2^n m^5) time. For example, for n=3, I could calculate 5..8 * 2^m, so let's say: 5, 6, 7, 8, 10, 12, 14, 16, 20, 24, 28, 32, 40, 48, 56, 64, ... 5*2^m, 6*2^m, 7*2^m, 8*2^m This would allow me to more accurately estimate a_1, because I would be able to monitor the peaks and troughs of the oscillations in the relative error. So hopefully, within a week or two, I should have an estimate of a_1 to at least 1-2 more digits than the current 1.083063. Who knows, I might soon have the most accurate estimate? Short of a closed form to calculate it, the only way I know to determine its value accurately is calculate very large terms in the sequence, isolate a local minimum and local maximum, and estimate the midpoint. BTW, the locations of the local maxima and minima follow a recurrence similar to locations of the zeroes. I don't have the numbers in front of me to post, so I will try to find which spreadsheet I buried them in and post the details later. I'll be working at a factory location today, so it'll probably be at least a day or two before I can update the group. ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread My favorite integer sequence - by tommy1729 - 07/27/2014, 08:44 PM RE: My favorite integer sequence - by jaydfox - 07/31/2014, 01:51 AM RE: My favorite integer sequence - by jaydfox - 08/01/2014, 01:53 AM RE: My favorite integer sequence - by Gottfried - 08/01/2014, 03:25 PM RE: My favorite integer sequence - by jaydfox - 08/01/2014, 05:04 PM RE: My favorite integer sequence - by sheldonison - 08/01/2014, 11:36 PM RE: My favorite integer sequence - by jaydfox - 08/01/2014, 11:57 PM RE: My favorite integer sequence - by jaydfox - 08/05/2014, 04:49 PM RE: My favorite integer sequence - by jaydfox - 08/05/2014, 05:57 PM RE: My favorite integer sequence - by Gottfried - 08/06/2014, 04:38 PM RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:19 AM RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:34 AM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 09:42 AM RE: My favorite integer sequence - by Gottfried - 08/09/2014, 02:27 PM RE: My favorite integer sequence - by tommy1729 - 09/09/2014, 12:55 AM RE: My favorite integer sequence - by jaydfox - 08/08/2014, 12:55 AM RE: My favorite integer sequence - by Gottfried - 08/08/2014, 02:27 AM RE: My favorite integer sequence - by jaydfox - 09/09/2014, 07:43 PM RE: My favorite integer sequence - by jaydfox - 09/09/2014, 09:45 PM RE: My favorite integer sequence - by jaydfox - 08/02/2014, 12:08 AM RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:38 PM RE: My favorite integer sequence - by sheldonison - 08/04/2014, 11:49 PM RE: My favorite integer sequence - by jaydfox - 09/16/2014, 05:32 AM RE: My favorite integer sequence - by Gottfried - 09/17/2014, 07:39 PM RE: My favorite integer sequence - by jaydfox - 10/02/2014, 10:53 PM RE: My favorite integer sequence - by Gottfried - 08/03/2014, 03:32 PM RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:44 PM RE: My favorite integer sequence - by sheldonison - 08/02/2014, 05:48 AM RE: My favorite integer sequence - by tommy1729 - 09/10/2014, 08:57 PM RE: My favorite integer sequence - by Gottfried - 08/02/2014, 07:43 PM RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:29 PM RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:36 PM RE: My favorite integer sequence - by tommy1729 - 08/05/2014, 11:16 PM RE: My favorite integer sequence - by tommy1729 - 08/08/2014, 11:02 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 07:02 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:51 PM RE: My favorite integer sequence - by sheldonison - 08/11/2014, 04:51 PM RE: My favorite integer sequence - by jaydfox - 08/11/2014, 05:19 PM RE: My favorite integer sequence - by jaydfox - 08/19/2014, 01:36 AM RE: My favorite integer sequence - by jaydfox - 08/19/2014, 02:05 AM RE: My favorite integer sequence - by jaydfox - 08/19/2014, 05:31 PM RE: My favorite integer sequence - by sheldonison - 08/19/2014, 07:56 PM RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:42 AM RE: My favorite integer sequence - by sheldonison - 08/20/2014, 02:11 PM RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:57 PM RE: My favorite integer sequence - by jaydfox - 08/21/2014, 01:15 AM RE: My favorite integer sequence - by jaydfox - 08/21/2014, 05:25 AM RE: My favorite integer sequence - by jaydfox - 08/22/2014, 05:39 PM RE: My favorite integer sequence - by jaydfox - 09/11/2014, 01:33 AM RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 09:16 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:19 PM RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 10:52 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 11:46 PM RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 11:10 PM RE: My favorite integer sequence - by jaydfox - 08/10/2014, 12:30 AM RE: My favorite integer sequence - by tommy1729 - 08/11/2014, 12:17 PM RE: My favorite integer sequence - by Gottfried - 08/22/2014, 12:30 AM Amazing variant - by tommy1729 - 08/26/2014, 08:57 PM RE: My favorite integer sequence - by tommy1729 - 09/01/2014, 10:37 PM RE: My favorite integer sequence - by tommy1729 - 10/02/2014, 11:24 PM RE: My favorite integer sequence - by jaydfox - 10/02/2014, 11:29 PM RE: My favorite integer sequence - by tommy1729 - 02/10/2015, 12:15 AM RE: My favorite integer sequence - by tommy1729 - 02/15/2015, 05:19 PM RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 08:22 AM RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 09:10 PM RE: My favorite integer sequence - by tommy1729 - 03/13/2016, 12:31 AM

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