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Does anyone have any idea how to solve this recurrence relation in the general case?

$$f_{x,y}=-f_{{x-1},{y-1}}f_{{x},{y-1}}f_{{x+1},{y-1}}-f_{{x},{y-1}}f_{{x+1},{y-1}}+f_{x,{y-1}}+f_{{x+1},{y-1}}$$

I can solve the relation corresponding to each summand ($f_{x,y}=-f_{{x-1},{y-1}}f_{{x},{y-1}}f_{{x+1},{y-1}}$), but I don't see that those solutions can be combined in any obvious way to create a solution for the whole relation.

Jean Marie
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alspmrg
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  • I don't agree to close this question which is well formulated with an attempt of solution. – Jean Marie May 01 '20 at 07:50
  • Which initial conditions/values do you take ? – Jean Marie May 01 '20 at 07:56
  • Two considerations : 1) Something puzzles me : how can you be sure that a solution exists because your relationship, as it is written, "defines" (in the LHS) $f_{x,y}$ through a formula involving in its RHS "forward terms" in $x+1$ 2) An efficient method for solving recurrence relationships with 2 variables is to use generating functions, such as described in the solution here. – Jean Marie May 01 '20 at 08:06
  • Can you say something about the context in which you are interested by this relationship ? – Jean Marie May 01 '20 at 11:01
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    This recurrence relation arises in trying to find a function which models the cellular automaton rule 110. As for initial conditions, I guess $f_{00}=1$, $f_{0y}=0$, $y \neq 0$ would be interesting. – alspmrg May 01 '20 at 14:53
  • I see, the equivalent of a boolean formula like Mod[(1+p)qr+q+r,2] in https://www.wolframscience.com/nks/notes-2-1--cellular-automaton-rules-as-formulas/ – Jean Marie May 01 '20 at 15:45
  • Interesting things in the Google book version of "Irreducibility and Computational Equivalence 10 Years After Wolfram's A New Kind of Science" Editors: Zenil, Hector, Springer, 2013, beginning at page 249 – Jean Marie May 01 '20 at 15:55

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