# `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 further parameters). The result is pretty printed. For example,

```read("recurrence-guess.gp");
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    roots 2.00000
x - 1        roots 1.00000

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

as powers
n * 2^n
+ 1

OEIS
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4).
%F a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).```

The guess is found by a simple `matsolve()`. Linear recurrences include powers, polynomials, and polynomials times powers. Values given can themselves be GP polynomials for parameterization, or (very) limited symbolic calculation, or bivariate gfs guessed on one variable.

`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 16 here. Requires my `pol-pfrac.gp`.

`recurrence-guess.gp` (53k, and sig)
`recurrence-guess-16.tar.gz` (53k, and sig)

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

The `tar` file includes some self-tests, and the following `examples/polmod.gp` script illustrating linear recurrence evaluation using `t_POLMOD`, which is efficient and compact but a little obscure.

`polmod.gp` (18k, and sig)

## Install

To install so `recurrence_guess()` is always available interactively, put `recurrence-guess.gp` in say your `~/gp` directory (which is in the GP `default(path)`) then in file `~/.gprc`

`read "recurrence-guess.gp"`
Give a full path (possibly starting `~/`) if installed somewhere else.

## Other Ways to Do It

Similar code can be found in

Bill Allombert points out too that `bestapprPade(Ser(vec))` gives a generating function. On a long recurrence, sometimes `lindep()` seems much faster than `matsolve()` (would intend to use that if so). The nice output is the tedious part. Of course "nice" is a matter of personal preference and the output is still quite mechanical.

This page Copyright 2016, 2017, 2018, 2019, 2020, 2021, 2024 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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