Sequences defined/computed by polynomial recurrence

[Colin]

Hi, I'm looking to compile a list of sequences that can be defined by a(n) = P(a(n-1)) for some polynomial function P(x). I have derived an extension of sequences defined by that for various cases of P(x) to the reals and am realizing I have very examples for my paper (also I'm curious what the implications are if it even makes any sense). The only non-trivial example I have is A086714 which doesn't have anything interesting. Does anyone else have any other examples?

I'm going to post the extensions I found as soon as I finish writing my paper. 

[Daniel Mondot]

It may not be what you are looking for, but a lot of sequences have a formula of the type:

a(n) = c + c1*a(n-1) + c2*a(n-2) + c3*a(n-3) + c4*a(n-4) + ... etc ...

Examples: A001906, A015523, A215007, A027934, A061646, A092521, A106521, A101427, A269447

Daniel.

On Tue, Oct 28, 2025 at 10:12 AM Colin <colin....@gmail.com> wrote:

Hi, I'm looking to compile a list of sequences that can be defined by a(n) = P(a(n-1)) for some polynomial function P(x). I have derived an extension of sequences defined by that for various cases of P(x) to the reals and am realizing I have very examples for my paper (also I'm curious what the implications are if it even makes any sense). The only non-trivial example I have is A086714 which doesn't have anything interesting. Does anyone else have any other examples?

I'm going to post the extensions I found as soon as I finish writing my paper. 

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[Jeffrey Shallit]

You should look at "polynomial iteration", "iterated maps", "iterates of polynomials", "polynomial maps", "polynomial recursions",  in the literature.  There are literally dozens of papers on this subject.   For example

https://www.numdam.org/item/CM_1996__103_3_319_0.pdf

https://academic.oup.com/plms/article-abstract/s3-68/2/225/1471442

https://arxiv.org/abs/2004.09353

https://link.springer.com/article/10.1007/s11139-021-00428-5

just to name a few.

On Tue, Oct 28, 2025 at 11:12 AM Colin <colin....@gmail.com> wrote:

Hi, I'm looking to compile a list of sequences that can be defined by a(n) = P(a(n-1)) for some polynomial function P(x). I have derived an extension of sequences defined by that for various cases of P(x) to the reals and am realizing I have very examples for my paper (also I'm curious what the implications are if it even makes any sense). The only non-trivial example I have is A086714 which doesn't have anything interesting. Does anyone else have any other examples?

I'm going to post the extensions I found as soon as I finish writing my paper. 

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