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,

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

Recurrence length=3
  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

  %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. is free software (free as in freedom), published under the terms of the GNU General Public License (v3 or higher). Download version 14 here. Requires my (51k, and sig)
recurrence-guess-14.tar.gz (51k, and sig)

Just and 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 an examples/ script illustrating linear recurrence evaluation using t_POLMOD, which is efficient and compact but a little obscure.

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

read ""
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 faster (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.

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