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 My favorite integer sequence Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 08/01/2014, 03:25 PM (This post was last modified: 08/03/2014, 10:21 AM by Gottfried.) I've got the generating rule in terms of a matrix. Assume an (ideally infinite sized) matrix with a simple generating-scheme, whose top-left segments is     Then the sequence A018819 occurs by taking M to infinite powers. Because the diagonal is zero except of the top left element M is nilpotent and the powers of M tend to become a column-vector in the first column. See for instance M^2:     and a higher power M^6:     Usually I try such approaches to consider diagonalization or Jordan-forms et al - perhaps for a suggestive pattern, for instance, A is an eigensequence for M because     However I did not yet find more interesting things in this manner. Interestingly the sequence generated by Jay's function $f(x)$ for real valued approximations to the elements of A has a vanishing binomial-transform: if that sequence of coefficients is premultiplied by the matrix P^-1 (the inverse of the lower triangular Pascalmatrix) then the transformed coefficients vanish quickly: (from left to right Jay's coefficients, the binomial-transforms, the original coefficients, and the binomial-transforms of the original coefficients):     One more observation which might be interesting. Consider the dotproduct of a Vandermondevector V(x) with M (where V(x) is a vector of consecutive powers of x such that $V(x)=[1,x,x^2,x^2,x^3,...]$). Then the dot-product $V(x)*M=Y$ gives a rowvector Y whose entries evaluate to geometric series such that $V(x)*M = 1/(1-x) * [1,x^2,x^4,x^6,...] = 1/(1-x)*V(x^2)$. Clearly this can be iterated: $V(x^2)*M = 1/(1-x^2) * [1,x^4,x^8,x^{12},...] = 1/(1-x^2)*V(x^4)$ and expressed with the power of M $V(x)*M = 1/(1-x) * V(x^{2^1})$. $V(x)*M^2 = 1/(1-x)*1/(1-x^2) * V(x^{2^2})$. ... $V(x)*M^h = 1/(1-x)*1/(1-x^2)*...*1/(1-x^{2^h}) * V(x^{2^{h+1}})$ . In the limit to infinite powers of M this gives for the first column in the result the scalar value $...$ $y = w(x) = 1 / (1-x) / (1-x ^ 2) / (1 - x ^ 4 ) \ldots$ and all others columns tend to zero. We might say, that in the above notation w(x) is the generating function for the sequence A. update: I changed the name of the function to not to interfer with Jay's function f(x) which interpolates the sequence A by real values of f(x) The *value* y, on the other hand, is then the evaluation of the power series whose coefficients are the terms of the original sequence at x: $y = w(x) = 1 + 1x + 2x^2 + 2x^3 + 4x^4+4x^5+...$ which seems to be convergent for |x|<1. I'm fiddling a bit more with it but do not yet expect much exciting news... Gottfried Gottfried Helms, Kassel « 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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