On Sunday, August 26, 2018 at 2:35:13 PM UTC-5, Richard Damon wrote: > On 8/26/18 1:58 PM, Musatov wrote: > > On Sunday, August 26, 2018 at 12:49:16 PM UTC-5, Richard Damon wrote: > >> On 8/26/18 12:48 PM, Dennis Lee Bieber wrote: > >>>> The sequence is defined by: > >>>> > >>>> For 1 <= n <= 3, a(n) = n; thereafter, a(2n) = a(n) + a(n+1), a(2n-1) = > >>>> a(n) + a(n-2). > <snip> > >>> I am not sure what 'fractal' property this sequence has that he > >>> wants to > >> display. > > I'm sorry, let me try to explain: > > > > Here is my output: > > 1, 2, 3, 5, 4, 8, 7, 9, 7, 12, 13, 15, 11, 16, 17, 16, 14, 19, 21, 25, 20, > > 28, 27, 26, 24, 27, 31, 33, 28, 33, 32, 30, 31, 33, 35, 40, 35, 46, 44, 45, > > 41, 48, 53, 55, 47, 53, 54, 50, 51, 51, 53, > > > > It is an OEIS sequence. > > > > I was told this image of the scatterplot emphasizes the 'fractal nature' of > > my sequence: > > > > https://oeis.org/A292575/a292575.png > > Something is wrong with that image compared to the sequence, as the > sequence is always positive, and in fact the lowest the sequence can get > to is always increasing (as it starts always positive, and each term is > the sum of two previous terms),while the graph is going negative. > > (actually going to the definition of the sequence, the plot isn't of > a(n) but a(n)-n, which can go negative) > > I normally think for fractals as a sequence of patterns of increasing > complexity, or a pattern looked at with increasing resolution revealing > the growth pattern. This sequence isn't quite like that, but I suppose > if you think of the sequence a(n) in the interval m <= n <= 2*m, and > then the interval 2*m <= n <= 4*m, that second interval is somewhat like > the first with some recursively added pattern (especially if you include > the -n in the sequence). > > That graph is probably the best way to show that pattern. > > One thing that might help, is to clean up the definition of a(n) to be > more directly computable, and maybe even include the subtraction of n. > > A rewriting of your rules would be: > > a(n) > > n=1,2,3: a(n) = n > > n>3, and even: a(n) = a(n/2) + a(n/2+1) > > n>3 and odd: a(n) = a((n+1)/2) + a(n-3)/2) > > If I have done my math right, this is the same sequence definition, but > always defining what a(n) is equal to. > > If we want to define the sequence b(n) = a(n) - n, we can transform the > above by substitution > > b(n) > > n=1,2,3: b(n) = 0 > > n>3 and even: b(n) = a(n/2)+a(n/2+1)-n > > = b(n/2)+b(n/2+1) + n/2 + n/2+1 -n > > = b(n/2) + b(n/2+1) + 1 > > n>3 and odd: b(n) = a((n+1)/2) + a((n-3)/2) - n > > = b((n+1)/2) + b((n-3)/2) + (n+1)/2 + (n-3)/2 -n > > = b((n+1)/2) + b((n-3)/2) -1 > > -- > Richard Damon
Thank you, Richard. If anyone is interested further, even in writing a Python code to generate the sequence or further preparing of an animation I would be delighted. Musatov -- https://mail.python.org/mailman/listinfo/python-list
