 # `recurrence-guess.gp`

This is a spot of Pari/GP code to guess a linear recurrence from a vector of numbers (or vector of polynomials for guessing with a symbolic parameter). If successful the result is pretty printed. For example,

```recurrence_guess([1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609]);
=>

Recurrence length=3
Coefficients
v[n-3]* [4, -8, 5] *v[n-1]  = v[n]
v[n] =  v[n-1]* [5, -8, 4] *v[n-3]

Characteristic polynomial
x^3 - 5*x^2 + 8*x - 4
= factors
(x - 2)^2  root 2.00000
x - 1      root 1.00000

Generating function
(1 - 2*x + 2*x^2)/(1 - 5*x + 8*x^2 - 4*x^3)
= partial fractions
1/(1 - x)
- 1/(1 - 2*x)
+ 1/(1 - 2*x)^2

As powers
n * 2^n
+ 1```

The guess is found by a simple `matsolve()`. Linear recurrences include powers, polynomials, and polynomials times powers. Similar code can be found in `findrec()` by Charles R. Greathouse and `ggf()` in the OEIS Sequence Tools. Bill Allombert points out too that `bestapprPade(Ser(vec))` gives a generating function. The nice display here in various forms is the tedious part. Of course "nice" is a matter of personal preference and the output is still quite mechanical.

`recurrence-guess.gp` is free software (free as in freedom), published under the terms of the GNU General Public License (v3 or higher). Download version 9 here. Requires my `pol-pfrac.gp`.

`recurrence-guess.gp` (45k, and sig)
`recurrence-guess-9.tar.gz` (42k, and sig)

Just `recurrence-guess.gp` and `pol-pfrac.gp` are enough to run. The `tar` file includes some self-tests. The sig files are Gnu PG ascii armoured signatures generated from my key.

This page Copyright 2016, 2017, 2018, 2019 Kevin Ryde, except for the GPLv3 logo which is Copyright Free Software Foundation and used here in accordance with its terms.

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