|
|
|
|
|
|
|
|
|
#45 by Vaclav Kotesovec at Mon Mar 13 09:11:48 EDT 2023
|
|
|
|
|
|
|
|
#44 by Michel Marcus at Fri Feb 24 01:49:52 EST 2023
|
|
|
|
|
|
Discussion
|
| Fri Feb 24
| 07:01
| Vaclav Kotesovec: You are right that it is more about upper and lower bounds.
|
| Sat Feb 25
| 16:31
| Jerzy R Borysowicz: Thank you. I am looking at zeros of b(n), they are easier to find than maxima. If zk is zero of b, b(zk)=0, then I think that zk =~ C*sqrt(k)
|
|
|
|
|
|
|
#43 by Michel Marcus at Fri Feb 24 01:49:35 EST 2023
|
|
| LINKS
|
<a href="/index/Com#comp">Index entries for sequences related to compositions</a>
<a href="/index/Com#comp">Index entries for sequences related to compositions</a>
|
|
| EXAMPLE
|
[ [1] [5]
[ [2] [3, 1, 1]
[ [3] [1, 3, 1]
[ [4] [1, 1, 3]
[ [5] [2, 2, 1]
[ [6] [2, 1, 2]
[ [7] [1, 2, 2]
|
|
| STATUS
|
proposed
editing
|
|
|
|
|
|
|
#42 by Jerzy R Borysowicz at Thu Feb 23 22:06:11 EST 2023
|
|
|
|
|
|
|
|
#41 by Jerzy R Borysowicz at Thu Feb 23 21:29:18 EST 2023
|
|
| FORMULA
|
Conjecture: abs(b(n)-1) < 0.0015015, where b(n) = a(n)*sqrt(n)/2^(n-1), for n > 781; b(n) does not have a limit. - Jerzy R Borysowicz, Feb 17 2023
|
|
| STATUS
|
proposed
editing
|
|
|
|
|
Discussion
|
| Thu Feb 23
| 22:05
| Jerzy R Borysowicz: b(n) looks a bit as the sin function(but it is not). At n=800000 the "period" (the distance between the local maxima(minima) ) is approximately 1812. The maximum 1.014384... is at 800126 and the minimum .985616... is at 801022. The "period" is approximately 2.3*sqrt(n), it is 1812
at n=800 000 and 220 at n=10050. The values of local maxima(minima) change very slowly at n>=800000 ( maybe they have a limit). Thank you for your comment. I will study "attractors" to see if it is a better description than in terms of maxima and minima. Does the sin function has attractors +-1 ?
|
|
|
|
|
|
|
#40 by Vaclav Kotesovec at Wed Feb 22 07:14:10 EST 2023
|
|
|
|
|
|
Discussion
|
| Wed Feb 22
| 11:03
| Vaclav Kotesovec: Numerically (from 100000 terms) are these two attractors 1.01438... and 0.98561...
|
|
|
|
|
|
|
#39 by Vaclav Kotesovec at Wed Feb 22 07:11:17 EST 2023
|
|
| LINKS
|
Vaclav Kotesovec, <a href="/A280352/a280352.jpg">Plot of a(n) / (2^(n-1)/sqrt(n)) for n = 1..10000</a>
|
|
| STATUS
|
proposed
editing
|
|
|
|
|
Discussion
|
| Wed Feb 22
| 07:14
| Vaclav Kotesovec: Yes, there is no limit, but there are two attractors, see the graph. But I think it should be 0.015 instead of 0.0015.
|
|
|
|
|
|
|
#38 by Jon E. Schoenfield at Fri Feb 17 18:48:25 EST 2023
|
|
|
|
|
|
|
|
#37 by Jon E. Schoenfield at Fri Feb 17 18:48:23 EST 2023
|
|
| FORMULA
|
G.f.: Sum_{k>=1} (x/(1 - -x))^(k*(k+1)/2).
a(n) = Sum{_{k=1..floor((sqrt(8*n+1)-1)/2)} binomial(n-1,, k*(k+1)/2-1). - Jerzy R Borysowicz, Dec 26 2022
Conjecture: abs(b(n)-1) < 0.0015, where b(n)=) = a(n)*sqrt(n)/2^(n-1), for n> > 781; b(n) does not have a limit. - Jerzy R Borysowicz, Feb 17 2023
|
|
| STATUS
|
proposed
editing
|
|
|
|
|
|
|
#36 by Jerzy R Borysowicz at Fri Feb 17 15:26:51 EST 2023
|
|
|
|
|
|
Discussion
|
| Fri Feb 17
| 15:28
| Jerzy R Borysowicz: ....... the asymptotic behaviour of a(n)
|
|
|
|
|
|
