PROG
(MAGMAMagma) I:=[0, 1, 2]; [n le 3 select I[n] else Self(n-1)*Self(n-2) - Self(n-3): n in [1..15]]; // G. C. Greubel, Mar 01 2018
Discussion
Thu Sep 08
08:44
OEIS Server: https://oeis.org/edit/global/2944
MATHEMATICA
a[01] = 0; a[12] = 1; a[23] = 2; a[n_] := a[n] = a[n - 1] a[n - 2] - a[n - 3]; Table[a[n], {n, 0, 1, 15}] (* Alonso del Arte, Jan 31 2011 *)
Discussion
Sat May 11
23:39
Petros Hadjicostas: Good, thank you.
23:47
Petros Hadjicostas: Jon, as I mentioned in another sequence, mathematician or computer scientist Hone has found the limit for many similar sequences (click on the sequences with Cf., and you will find a number of his contributions). Essentially, he ignored the constant c you found above. We have log(a(n))/phi^n ~ c, so log(log(a(n)) - n log(phi) ~ c, so (1/n)*log(log(a(n)) - log (phi) ~ (log c)/n, which implies lim_{n -> infinity} (1/n)*log(log(a(n)) = log(phi). He has many formulas like that where phi is replaced by solutions to equations of the form x^k - x^{k-1} -...-x-1=0. We can get better estimates than him by not killing the constant c.
23:57
Petros Hadjicostas: Jon, whenever you find time, please try your acceleration technique with sequence A061021. You will find another constant c.
Sun May 12
00:11
Jon E. Schoenfield: Thanks ... I'll try to have a look either tonight or tomorrow.
00:39
Petros Hadjicostas: Thanks! Consider your techniques on sequences A061021, A061292, A072877, A072878, A072880, and A074394. I am not sure if they also work for the Somos sequences A072876, A072875, etc.
FORMULA
It appears that lim_{n->infinity} log(a(n))/phi^n = 0.07743008049000107520747623421744398272089261907514..., where phi = (1 + sqrt(5))/2 is the golden ratio A001622. - Petros Hadjicostas and Jon E. Schoenfield, May 11 2019, May 11 2019
Discussion
Sat May 11
23:01
Jon E. Schoenfield: Um ... this. :-)
FORMULA
Conjecture: It appears that lim_{n->infinity} log(a(n)) ~ /phi^(n-1)* = 0.1252807743008049000107520747623421744398272089261907514... for large n, , where phi = (1 + sqrt(5))/2 is the golden ratio A001622. - Petros Hadjicostas, and _Jon E. Schoenfield_, May 11 2019, May 11 2019
Discussion
Sat May 11
23:00
Jon E. Schoenfield: How's this look?
Discussion
Sat May 11
20:34
Jon E. Schoenfield: Yes, you're right. Thanks for the correction! :-)
If we define t(n) = log(a(n))/phi^n for n > 1, then the real-valued sequence {t(2), t(3), t(4), t(5), ...} converges on the limit, but not as quickly as we might like. However, I found that the convergence can be greatly accelerated if we define
u(n) = (t(n)*(5 + sqrt(5)) + t(n-1)*(5 - sqrt(5)))/10
Doing so, I get
...
u(6) = 0.0832826420910264779679779931507623752615...
u(7) = 0.0787387587080768905273833014079867296063...
u(8) = 0.0775379275900890845976810693014742764906...
u(9) = 0.0774344538077412373616121159005381026076...
u(10) = 0.0774301076665953812948359123939708753190...
u(11) = 0.0774300805022735521898135220008213846618...
u(12) = 0.0774300804900011318292745634503184620858...
u(13) = 0.0774300804900010752074764253224400676219...
u(14) = 0.0774300804900010752074762342174439827257...
u(15) = 0.0774300804900010752074762342174439827208...
u(16) = 0.0774300804900010752074762342174439827208...
u(17) = 0.0774300804900010752074762342174439827208...
u(17) actually seems to give a result that's correct to about 150 digits.
20:44
Jon E. Schoenfield: @Petros -- Is this a good way to state the limit?
Lim_{n->infinity} log(a(n))/phi^n = 0.07743008049000107520747623421744398272089261907514...
Also, I can't remember if there's a suggested number of digits to include.... ?:-/
And I don't know whether it really needs to be stated as a conjecture, given the rapid acceleration. Does it?
Also, I feel bad about replacing your contribution with something from my computational results; I wouldn't have looked into the limiting behavior of this sequence at all, or known to look for something of the form log(a(n))/phi^(n-1) (or of the form log(a(n))/phi^n) if it hadn't been for your contribution. :-(
20:44
Jon E. Schoenfield: I really think you deserve most of the credit.
21:04
Petros Hadjicostas: Hi Jon: it is still a conjecture unless we give a rigorous proof. I do not know how many digits OEIS suggests. If you wish, please replace my contribution with yours and write: Petros Hadjicostas and Jon E. Schoenfield. (That will be the best!) By the way, check A072880, A072879, A072878, A072877, A072876, etc. You will be entering the "dangerous" domain of Somos sequences!!! There is some mathematician/computer scientist with the name Hone who apparently has a lot of such results (more powerful than ours!).