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 My favorite integer sequence jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/08/2014, 12:55 AM (08/06/2014, 04:38 PM)Gottfried Wrote: I'll also be much interested if I can learn about the approach how to determine the function f(x) which prognoses/approximates the elements of the sequence. I could implement the function f(x) in Pari/Gp but just did not get the clue how one would arrive at it. Gottfried Deriving the continuous function is actually pretty straightforward. I'll use exp(x) as an example: $ \exp(x) = \sum_{k=0}^{\infty} a_k x^k \\ a_0 = 1 \\ D\left[\exp(x)\right] = \exp(x)$ From this, you can derive exp(x). $ D\left[a_1 x\right] = a_0 \\ \Rightarrow a_1 = 1$ $ D\left[a_2 x^2\right] = a_1 x \\ \Rightarrow 2 a_2 x = a_1 x \\ \Rightarrow a_2 = 1/2$ $ D\left[a_3 x^3\right] = a_2 x^2 \\ \Rightarrow 3 a_3 x^2 = a_2 x^2 \\ \Rightarrow a_3 = \frac{1}{3}\left(\frac{1}{2}\right) = \frac{1}{6}$ In general: $ D\left[a_k x^k\right] = a_{k-1} x^{k-1} \\ \Rightarrow k a_k x^{k-1} = a_{k-1} x^{k-1} \\ \Rightarrow a_k = \frac{1}{k} \left(a_{k-1}\right)$ By induction: $ a_k = \frac{1}{k!}$ Now let's try with the problem at hand: $ f(x) = \sum_{k=0}^{\infty} b_k x^k \\ b_0 = 1 \\ D\left[f(x)\right] = f(x/2)$ From this, we start recursively solving: $ D\left[b_1 x\right] = b_0 \\ \Rightarrow b_1 = 1$ $ D\left[b_2 x^2\right] = b_1 \left(\frac{x}{2}\right) \\ \Rightarrow 2 b_2 x = \frac{1}{2} b_1 x \\ \Rightarrow b_2 = \frac{1}{2} \left(\frac{1}{2}\right) = \frac{1}{4}$ $ D\left[b_3 x^3\right] = b_2 \left(\frac{x}{2}\right)^2 \\ \Rightarrow 3 b_3 x^2 = \frac{1}{2^2} b_2 x^2 \\ \Rightarrow b_3 = \frac{1}{3} \left(\frac{1}{2^2}\right) \left(\frac{1}{4}\right) = \frac{1}{48}$ In general: $ D\left[b_k x^k\right] = b_{k-1} \left(\frac{x}{2}\right)^{k-1} \\ \Rightarrow k b_k x^{k-1} = \frac{1}{2^{k-1}} b_{k-1} x^{k-1} \\ \Rightarrow b_k = \frac{1}{k} \left(\frac{1}{2^{k-1}}\right) b_{k-1}$ By induction: $ b_k = \frac{1}{2^{(k-1)k/2} k!}$ Then, as I posted previously: $ f(x) = \sum_{k=0}^{\infty}\frac{1}{2^{k(k-1)/2} k!} x^k$ ~ 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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