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Help with linear recurrence in 4 variables

Mario Weitzer
Posted 11 years ago

Hello everybody!

I would need help with solving the following linear recurrence (finding a closed form):

r(a,b,c,d) = (r(a-1,b+1,c+1,d-1) * a * (4z-b-2c-4d+14) + r(a,b,c+1,d-1) * (2a+b+c+4) * (-2z+b+c+2d-7) + r(a,b+2,c,d-1) * (-z+a+b+c+d) * (-2z+c+2d-7) + r(a+1,b,c+1,d-1) * (-z+a+b+c+d) * (4z-2b-2c-4d+14)) / ((-2z+c+2d-7) * (2z-d+9))

r(a,b,c,0) = f(a,b,c)

where z is a non-negative integer and a+b+c+d <= z and f(a,b,c) is some known function. Of course if you iteratively apply r(a,b,c,d) d times you get down to r(a,b,c,0) = f(a,b,c) but so far I was unable to find any closed form both automatically with Mathematica and by hand. Does anybody have an idea how it could be done? I would really appreciate any hint!
POSTED BY: Mario Weitzer
5 Replies
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What is the simplest form of f(a,b,c) still making sense to you? What about z = 4 so there are only 2^4 states or argument sets of r? You probably will not do it with undetermined z and undertermined f in the first guesses.
POSTED BY: Udo Krause
Mario Weitzer
Mario Weitzer
Posted 11 years ago

f(a,b,c) is a very complicated expression and z can be any integer.
POSTED BY: Mario Weitzer

Because the condition a+b+c+d<=z is implemented in the definition of r, the number and type of the appearing f terms depend on z for fixed set a,b,c,d

In[54]:= Union[Variables[Simplify[r[4, 3, 5, 9, #]]]] & /@ 
 Range[21, 31]

Out[54]= {{f[0, 7, 14], f[1, 6, 14], f[2, 5, 14], f[3, 4, 14], 
  f[4, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], f[1, 6, 14], 
  f[1, 7, 14], f[1, 8, 13], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], 
  f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], f[4, 3, 14], f[4, 4, 14], 
  f[4, 5, 13], f[5, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], 
  f[0, 9, 14], f[0, 10, 13], f[0, 11, 12], f[1, 6, 14], f[1, 7, 14], 
  f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], f[1, 10, 12], f[2, 5, 14], 
  f[2, 6, 14], f[2, 7, 13], f[2, 7, 14], f[2, 8, 13], f[2, 9, 12], 
  f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], f[3, 6, 14], f[3, 7, 13], 
  f[3, 8, 12], f[4, 3, 14], f[4, 4, 14], f[4, 5, 13], f[4, 5, 14], 
  f[4, 6, 13], f[4, 7, 12], f[5, 3, 14], f[5, 4, 14], f[5, 5, 13], 
  f[6, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], f[0, 9, 14], 
  f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], f[0, 12, 12], 
  f[0, 13, 11], f[1, 6, 14], f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], 
  f[1, 9, 13], f[1, 9, 14], f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], 
  f[1, 12, 11], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], f[2, 7, 14], 
  f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], f[2, 10, 12], 
  f[2, 11, 11], f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], f[3, 6, 14], 
  f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], f[3, 8, 13], f[3, 9, 12], 
  f[3, 10, 11], f[4, 3, 14], f[4, 4, 14], f[4, 5, 13], f[4, 5, 14], 
  f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], f[4, 7, 13], f[4, 8, 12], 
  f[4, 9, 11], f[5, 3, 14], f[5, 4, 14], f[5, 5, 13], f[5, 5, 14], 
  f[5, 6, 13], f[5, 7, 12], f[6, 3, 14], f[6, 4, 14], f[6, 5, 13], 
  f[7, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], f[0, 9, 14], 
  f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], f[0, 12, 12], 
  f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[1, 6, 14], f[1, 7, 14], 
  f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], f[1, 9, 14], f[1, 10, 12], 
  f[1, 10, 13], f[1, 11, 12], f[1, 12, 11], f[1, 12, 12], 
  f[1, 13, 11], f[1, 14, 10], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], 
  f[2, 7, 14], f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], 
  f[2, 10, 12], f[2, 10, 13], f[2, 11, 11], f[2, 11, 12], 
  f[2, 12, 11], f[2, 13, 10], f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], 
  f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], f[3, 8, 13], 
  f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], f[3, 10, 12], 
  f[3, 11, 11], f[3, 12, 10], f[4, 3, 14], f[4, 4, 14], f[4, 5, 13], 
  f[4, 5, 14], f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], f[4, 7, 13], 
  f[4, 7, 14], f[4, 8, 12], f[4, 8, 13], f[4, 9, 11], f[4, 9, 12], 
  f[4, 10, 11], f[4, 11, 10], f[5, 3, 14], f[5, 4, 14], f[5, 5, 13], 
  f[5, 5, 14], f[5, 6, 13], f[5, 6, 14], f[5, 7, 12], f[5, 7, 13], 
  f[5, 8, 12], f[5, 9, 11], f[6, 3, 14], f[6, 4, 14], f[6, 5, 13], 
  f[6, 5, 14], f[6, 6, 13], f[6, 7, 12], f[7, 3, 14], f[7, 4, 14], 
  f[7, 5, 13], f[8, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], 
  f[0, 9, 14], f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], f[0, 12, 12],
   f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[0, 17, 9], f[1, 6, 14],
   f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], f[1, 9, 14], 
  f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], f[1, 12, 11], 
  f[1, 12, 12], f[1, 13, 11], f[1, 14, 10], f[1, 15, 10], f[1, 16, 9],
   f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], f[2, 7, 14], f[2, 8, 13], 
  f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], f[2, 10, 12], f[2, 10, 13], 
  f[2, 11, 11], f[2, 11, 12], f[2, 12, 11], f[2, 13, 10], 
  f[2, 13, 11], f[2, 14, 10], f[2, 15, 9], f[3, 4, 14], f[3, 5, 14], 
  f[3, 6, 13], f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], 
  f[3, 8, 13], f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], 
  f[3, 10, 12], f[3, 11, 11], f[3, 11, 12], f[3, 12, 10], 
  f[3, 12, 11], f[3, 13, 10], f[3, 14, 9], f[4, 3, 14], f[4, 4, 14], 
  f[4, 5, 13], f[4, 5, 14], f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], 
  f[4, 7, 13], f[4, 7, 14], f[4, 8, 12], f[4, 8, 13], f[4, 9, 11], 
  f[4, 9, 12], f[4, 9, 13], f[4, 10, 11], f[4, 10, 12], f[4, 11, 10], 
  f[4, 11, 11], f[4, 12, 10], f[4, 13, 9], f[5, 3, 14], f[5, 4, 14], 
  f[5, 5, 13], f[5, 5, 14], f[5, 6, 13], f[5, 6, 14], f[5, 7, 12], 
  f[5, 7, 13], f[5, 7, 14], f[5, 8, 12], f[5, 8, 13], f[5, 9, 11], 
  f[5, 9, 12], f[5, 10, 11], f[5, 11, 10], f[6, 3, 14], f[6, 4, 14], 
  f[6, 5, 13], f[6, 5, 14], f[6, 6, 13], f[6, 6, 14], f[6, 7, 12], 
  f[6, 7, 13], f[6, 8, 12], f[6, 9, 11], f[7, 3, 14], f[7, 4, 14], 
  f[7, 5, 13], f[7, 5, 14], f[7, 6, 13], f[7, 7, 12], f[8, 3, 14], 
  f[8, 4, 14], f[8, 5, 13], f[9, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], 
  f[0, 9, 13], f[0, 9, 14], f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], 
  f[0, 12, 12], f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[0, 17, 9],
   f[1, 6, 14], f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], 
  f[1, 9, 14], f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], f[1, 12, 11],
   f[1, 12, 12], f[1, 13, 11], f[1, 14, 10], f[1, 15, 10], 
  f[1, 16, 9], f[1, 18, 8], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], 
  f[2, 7, 14], f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], 
  f[2, 10, 12], f[2, 10, 13], f[2, 11, 11], f[2, 11, 12], 
  f[2, 12, 11], f[2, 13, 10], f[2, 13, 11], f[2, 14, 10], f[2, 15, 9],
   f[2, 16, 9], f[2, 17, 8], f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], 
  f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], f[3, 8, 13], 
  f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], f[3, 10, 12], 
  f[3, 11, 11], f[3, 11, 12], f[3, 12, 10], f[3, 12, 11], 
  f[3, 13, 10], f[3, 14, 9], f[3, 14, 10], f[3, 15, 9], f[3, 16, 8], 
  f[4, 3, 14], f[4, 4, 14], f[4, 5, 13], f[4, 5, 14], f[4, 6, 13], 
  f[4, 6, 14], f[4, 7, 12], f[4, 7, 13], f[4, 7, 14], f[4, 8, 12], 
  f[4, 8, 13], f[4, 9, 11], f[4, 9, 12], f[4, 9, 13], f[4, 10, 11], 
  f[4, 10, 12], f[4, 11, 10], f[4, 11, 11], f[4, 12, 10], 
  f[4, 12, 11], f[4, 13, 9], f[4, 13, 10], f[4, 14, 9], f[4, 15, 8], 
  f[5, 3, 14], f[5, 4, 14], f[5, 5, 13], f[5, 5, 14], f[5, 6, 13], 
  f[5, 6, 14], f[5, 7, 12], f[5, 7, 13], f[5, 7, 14], f[5, 8, 12], 
  f[5, 8, 13], f[5, 9, 11], f[5, 9, 12], f[5, 10, 11], f[5, 10, 12], 
  f[5, 11, 10], f[5, 11, 11], f[5, 12, 10], f[5, 13, 9], f[6, 3, 14], 
  f[6, 4, 14], f[6, 5, 13], f[6, 5, 14], f[6, 6, 13], f[6, 6, 14], 
  f[6, 7, 12], f[6, 7, 13], f[6, 8, 12], f[6, 8, 13], f[6, 9, 11], 
  f[6, 9, 12], f[6, 10, 11], f[6, 11, 10], f[7, 3, 14], f[7, 4, 14], 
  f[7, 5, 13], f[7, 5, 14], f[7, 6, 13], f[7, 6, 14], f[7, 7, 12], 
  f[7, 7, 13], f[7, 8, 12], f[7, 9, 11], f[8, 3, 14], f[8, 4, 14], 
  f[8, 5, 13], f[8, 5, 14], f[8, 6, 13], f[8, 7, 12], f[9, 3, 14], 
  f[9, 4, 14], f[9, 5, 13], f[10, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], 
  f[0, 9, 13], f[0, 9, 14], f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], 
  f[0, 12, 12], f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[0, 17, 9],
   f[1, 6, 14], f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], 
  f[1, 9, 14], f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], f[1, 12, 11],
   f[1, 12, 12], f[1, 13, 11], f[1, 14, 10], f[1, 15, 10], 
  f[1, 16, 9], f[1, 18, 8], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], 
  f[2, 7, 14], f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], 
  f[2, 10, 12], f[2, 10, 13], f[2, 11, 11], f[2, 11, 12], 
  f[2, 12, 11], f[2, 13, 10], f[2, 13, 11], f[2, 14, 10], f[2, 15, 9],
   f[2, 16, 9], f[2, 17, 8], f[2, 19, 7], f[3, 4, 14], f[3, 5, 14], 
  f[3, 6, 13], f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], 
  f[3, 8, 13], f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], 
  f[3, 10, 12], f[3, 11, 11], f[3, 11, 12], f[3, 12, 10], 
  f[3, 12, 11], f[3, 13, 10], f[3, 14, 9], f[3, 14, 10], f[3, 15, 9], 
  f[3, 16, 8], f[3, 17, 8], f[3, 18, 7], f[4, 3, 14], f[4, 4, 14], 
  f[4, 5, 13], f[4, 5, 14], f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], 
  f[4, 7, 13], f[4, 7, 14], f[4, 8, 12], f[4, 8, 13], f[4, 9, 11], 
  f[4, 9, 12], f[4, 9, 13], f[4, 10, 11], f[4, 10, 12], f[4, 11, 10], 
  f[4, 11, 11], f[4, 12, 10], f[4, 12, 11], f[4, 13, 9], f[4, 13, 10],
   f[4, 14, 9], f[4, 15, 8], f[4, 15, 9], f[4, 16, 8], f[4, 17, 7], 
  f[5, 3, 14], f[5, 4, 14], f[5, 5, 13], f[5, 5, 14], f[5, 6, 13], 
  f[5, 6, 14], f[5, 7, 12], f[5, 7, 13], f[5, 7, 14], f[5, 8, 12], 
  f[5, 8, 13], f[5, 9, 11], f[5, 9, 12], f[5, 10, 11], f[5, 10, 12], 
  f[5, 11, 10], f[5, 11, 11], f[5, 12, 10], f[5, 13, 9], f[5, 13, 10],
   f[5, 14, 9], f[5, 15, 8], f[6, 3, 14], f[6, 4, 14], f[6, 5, 13], 
  f[6, 5, 14], f[6, 6, 13], f[6, 6, 14], f[6, 7, 12], f[6, 7, 13], 
  f[6, 8, 12], f[6, 8, 13], f[6, 9, 11], f[6, 9, 12], f[6, 10, 11], 
  f[6, 11, 10], f[6, 11, 11], f[6, 12, 10], f[6, 13, 9], f[7, 3, 14], 
  f[7, 4, 14], f[7, 5, 13], f[7, 5, 14], f[7, 6, 13], f[7, 6, 14], 
  f[7, 7, 12], f[7, 7, 13], f[7, 8, 12], f[7, 9, 11], f[7, 9, 12], 
  f[7, 10, 11], f[7, 11, 10], f[8, 3, 14], f[8, 4, 14], f[8, 5, 13], 
  f[8, 5, 14], f[8, 6, 13], f[8, 7, 12], f[8, 7, 13], f[8, 8, 12], 
  f[8, 9, 11], f[9, 3, 14], f[9, 4, 14], f[9, 5, 13], f[9, 5, 14], 
  f[9, 6, 13], f[9, 7, 12], f[10, 3, 14], f[10, 4, 14], f[10, 5, 13], 
  f[11, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], f[0, 9, 14], 
  f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], f[0, 12, 12], 
  f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[0, 17, 9], f[1, 6, 14], 
  f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], f[1, 9, 14], 
  f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], f[1, 12, 11], 
  f[1, 12, 12], f[1, 13, 11], f[1, 14, 10], f[1, 15, 10], f[1, 16, 9],
   f[1, 18, 8], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], f[2, 7, 14], 
  f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], f[2, 10, 12], 
  f[2, 10, 13], f[2, 11, 11], f[2, 11, 12], f[2, 12, 11], 
  f[2, 13, 10], f[2, 13, 11], f[2, 14, 10], f[2, 15, 9], f[2, 16, 9], 
  f[2, 17, 8], f[2, 19, 7], f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], 
  f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], f[3, 8, 13], 
  f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], f[3, 10, 12], 
  f[3, 11, 11], f[3, 11, 12], f[3, 12, 10], f[3, 12, 11], 
  f[3, 13, 10], f[3, 14, 9], f[3, 14, 10], f[3, 15, 9], f[3, 16, 8], 
  f[3, 17, 8], f[3, 18, 7], f[3, 20, 6], f[4, 3, 14], f[4, 4, 14], 
  f[4, 5, 13], f[4, 5, 14], f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], 
  f[4, 7, 13], f[4, 7, 14], f[4, 8, 12], f[4, 8, 13], f[4, 9, 11], 
  f[4, 9, 12], f[4, 9, 13], f[4, 10, 11], f[4, 10, 12], f[4, 11, 10], 
  f[4, 11, 11], f[4, 12, 10], f[4, 12, 11], f[4, 13, 9], f[4, 13, 10],
   f[4, 14, 9], f[4, 15, 8], f[4, 15, 9], f[4, 16, 8], f[4, 17, 7], 
  f[4, 18, 7], f[4, 19, 6], f[5, 3, 14], f[5, 4, 14], f[5, 5, 13], 
  f[5, 5, 14], f[5, 6, 13], f[5, 6, 14], f[5, 7, 12], f[5, 7, 13], 
  f[5, 7, 14], f[5, 8, 12], f[5, 8, 13], f[5, 9, 11], f[5, 9, 12], 
  f[5, 10, 11], f[5, 10, 12], f[5, 11, 10], f[5, 11, 11], 
  f[5, 12, 10], f[5, 13, 9], f[5, 13, 10], f[5, 14, 9], f[5, 15, 8], 
  f[5, 16, 8], f[5, 17, 7], f[6, 3, 14], f[6, 4, 14], f[6, 5, 13], 
  f[6, 5, 14], f[6, 6, 13], f[6, 6, 14], f[6, 7, 12], f[6, 7, 13], 
  f[6, 8, 12], f[6, 8, 13], f[6, 9, 11], f[6, 9, 12], f[6, 10, 11], 
  f[6, 11, 10], f[6, 11, 11], f[6, 12, 10], f[6, 13, 9], f[6, 14, 9], 
  f[6, 15, 8], f[7, 3, 14], f[7, 4, 14], f[7, 5, 13], f[7, 5, 14], 
  f[7, 6, 13], f[7, 6, 14], f[7, 7, 12], f[7, 7, 13], f[7, 8, 12], 
  f[7, 9, 11], f[7, 9, 12], f[7, 10, 11], f[7, 11, 10], f[7, 12, 10], 
  f[7, 13, 9], f[8, 3, 14], f[8, 4, 14], f[8, 5, 13], f[8, 5, 14], 
  f[8, 6, 13], f[8, 7, 12], f[8, 7, 13], f[8, 8, 12], f[8, 9, 11], 
  f[8, 10, 11], f[8, 11, 10], f[9, 3, 14], f[9, 4, 14], f[9, 5, 13], 
  f[9, 5, 14], f[9, 6, 13], f[9, 7, 12], f[9, 8, 12], f[9, 9, 11], 
  f[10, 3, 14], f[10, 4, 14], f[10, 5, 13], f[10, 6, 13], 
  f[10, 7, 12], f[11, 3, 14], f[11, 4, 14], f[11, 5, 13], 
  f[12, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], f[0, 9, 14], 
  f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], f[0, 12, 12], 
  f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[0, 17, 9], f[1, 6, 14], 
  f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], f[1, 9, 14], 
  f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], f[1, 12, 11], 
  f[1, 12, 12], f[1, 13, 11], f[1, 14, 10], f[1, 15, 10], f[1, 16, 9],
   f[1, 18, 8], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], f[2, 7, 14], 
  f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], f[2, 10, 12], 
  f[2, 10, 13], f[2, 11, 11], f[2, 11, 12], f[2, 12, 11], 
  f[2, 13, 10], f[2, 13, 11], f[2, 14, 10], f[2, 15, 9], f[2, 16, 9], 
  f[2, 17, 8], f[2, 19, 7], f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], 
  f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], f[3, 8, 13], 
  f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], f[3, 10, 12], 
  f[3, 11, 11], f[3, 11, 12], f[3, 12, 10], f[3, 12, 11], 
  f[3, 13, 10], f[3, 14, 9], f[3, 14, 10], f[3, 15, 9], f[3, 16, 8], 
  f[3, 17, 8], f[3, 18, 7], f[3, 20, 6], f[4, 3, 14], f[4, 4, 14], 
  f[4, 5, 13], f[4, 5, 14], f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], 
  f[4, 7, 13], f[4, 7, 14], f[4, 8, 12], f[4, 8, 13], f[4, 9, 11], 
  f[4, 9, 12], f[4, 9, 13], f[4, 10, 11], f[4, 10, 12], f[4, 11, 10], 
  f[4, 11, 11], f[4, 12, 10], f[4, 12, 11], f[4, 13, 9], f[4, 13, 10],
   f[4, 14, 9], f[4, 15, 8], f[4, 15, 9], f[4, 16, 8], f[4, 17, 7], 
  f[4, 18, 7], f[4, 19, 6], f[4, 21, 5], f[5, 3, 14], f[5, 4, 14], 
  f[5, 5, 13], f[5, 5, 14], f[5, 6, 13], f[5, 6, 14], f[5, 7, 12], 
  f[5, 7, 13], f[5, 7, 14], f[5, 8, 12], f[5, 8, 13], f[5, 9, 11], 
  f[5, 9, 12], f[5, 10, 11], f[5, 10, 12], f[5, 11, 10], f[5, 11, 11],
   f[5, 12, 10], f[5, 13, 9], f[5, 13, 10], f[5, 14, 9], f[5, 15, 8], 
  f[5, 16, 8], f[5, 17, 7], f[5, 19, 6], f[6, 3, 14], f[6, 4, 14], 
  f[6, 5, 13], f[6, 5, 14], f[6, 6, 13], f[6, 6, 14], f[6, 7, 12], 
  f[6, 7, 13], f[6, 8, 12], f[6, 8, 13], f[6, 9, 11], f[6, 9, 12], 
  f[6, 10, 11], f[6, 11, 10], f[6, 11, 11], f[6, 12, 10], f[6, 13, 9],
   f[6, 14, 9], f[6, 15, 8], f[6, 17, 7], f[7, 3, 14], f[7, 4, 14], 
  f[7, 5, 13], f[7, 5, 14], f[7, 6, 13], f[7, 6, 14], f[7, 7, 12], 
  f[7, 7, 13], f[7, 8, 12], f[7, 9, 11], f[7, 9, 12], f[7, 10, 11], 
  f[7, 11, 10], f[7, 12, 10], f[7, 13, 9], f[7, 15, 8], f[8, 3, 14], 
  f[8, 4, 14], f[8, 5, 13], f[8, 5, 14], f[8, 6, 13], f[8, 7, 12], 
  f[8, 7, 13], f[8, 8, 12], f[8, 9, 11], f[8, 10, 11], f[8, 11, 10], 
  f[8, 13, 9], f[9, 3, 14], f[9, 4, 14], f[9, 5, 13], f[9, 5, 14], 
  f[9, 6, 13], f[9, 7, 12], f[9, 8, 12], f[9, 9, 11], f[9, 11, 10], 
  f[10, 3, 14], f[10, 4, 14], f[10, 5, 13], f[10, 6, 13], 
  f[10, 7, 12], f[10, 9, 11], f[11, 3, 14], f[11, 4, 14], 
  f[11, 5, 13], f[11, 7, 12], f[12, 3, 14], f[12, 5, 13], 
  f[13, 3, 14]}, {f[0, 7, 14], f[0, 8, 14], f[0, 9, 13], f[0, 9, 14], 
  f[0, 10, 13], f[0, 11, 12], f[0, 11, 13], f[0, 12, 12], 
  f[0, 13, 11], f[0, 14, 11], f[0, 15, 10], f[0, 17, 9], f[1, 6, 14], 
  f[1, 7, 14], f[1, 8, 13], f[1, 8, 14], f[1, 9, 13], f[1, 9, 14], 
  f[1, 10, 12], f[1, 10, 13], f[1, 11, 12], f[1, 12, 11], 
  f[1, 12, 12], f[1, 13, 11], f[1, 14, 10], f[1, 15, 10], f[1, 16, 9],
   f[1, 18, 8], f[2, 5, 14], f[2, 6, 14], f[2, 7, 13], f[2, 7, 14], 
  f[2, 8, 13], f[2, 8, 14], f[2, 9, 12], f[2, 9, 13], f[2, 10, 12], 
  f[2, 10, 13], f[2, 11, 11], f[2, 11, 12], f[2, 12, 11], 
  f[2, 13, 10], f[2, 13, 11], f[2, 14, 10], f[2, 15, 9], f[2, 16, 9], 
  f[2, 17, 8], f[2, 19, 7], f[3, 4, 14], f[3, 5, 14], f[3, 6, 13], 
  f[3, 6, 14], f[3, 7, 13], f[3, 7, 14], f[3, 8, 12], f[3, 8, 13], 
  f[3, 8, 14], f[3, 9, 12], f[3, 9, 13], f[3, 10, 11], f[3, 10, 12], 
  f[3, 11, 11], f[3, 11, 12], f[3, 12, 10], f[3, 12, 11], 
  f[3, 13, 10], f[3, 14, 9], f[3, 14, 10], f[3, 15, 9], f[3, 16, 8], 
  f[3, 17, 8], f[3, 18, 7], f[3, 20, 6], f[4, 3, 14], f[4, 4, 14], 
  f[4, 5, 13], f[4, 5, 14], f[4, 6, 13], f[4, 6, 14], f[4, 7, 12], 
  f[4, 7, 13], f[4, 7, 14], f[4, 8, 12], f[4, 8, 13], f[4, 9, 11], 
  f[4, 9, 12], f[4, 9, 13], f[4, 10, 11], f[4, 10, 12], f[4, 11, 10], 
  f[4, 11, 11], f[4, 12, 10], f[4, 12, 11], f[4, 13, 9], f[4, 13, 10],
   f[4, 14, 9], f[4, 15, 8], f[4, 15, 9], f[4, 16, 8], f[4, 17, 7], 
  f[4, 18, 7], f[4, 19, 6], f[4, 21, 5], f[5, 3, 14], f[5, 4, 14], 
  f[5, 5, 13], f[5, 5, 14], f[5, 6, 13], f[5, 6, 14], f[5, 7, 12], 
  f[5, 7, 13], f[5, 7, 14], f[5, 8, 12], f[5, 8, 13], f[5, 9, 11], 
  f[5, 9, 12], f[5, 10, 11], f[5, 10, 12], f[5, 11, 10], f[5, 11, 11],
   f[5, 12, 10], f[5, 13, 9], f[5, 13, 10], f[5, 14, 9], f[5, 15, 8], 
  f[5, 16, 8], f[5, 17, 7], f[5, 19, 6], f[6, 3, 14], f[6, 4, 14], 
  f[6, 5, 13], f[6, 5, 14], f[6, 6, 13], f[6, 6, 14], f[6, 7, 12], 
  f[6, 7, 13], f[6, 8, 12], f[6, 8, 13], f[6, 9, 11], f[6, 9, 12], 
  f[6, 10, 11], f[6, 11, 10], f[6, 11, 11], f[6, 12, 10], f[6, 13, 9],
   f[6, 14, 9], f[6, 15, 8], f[6, 17, 7], f[7, 3, 14], f[7, 4, 14], 
  f[7, 5, 13], f[7, 5, 14], f[7, 6, 13], f[7, 6, 14], f[7, 7, 12], 
  f[7, 7, 13], f[7, 8, 12], f[7, 9, 11], f[7, 9, 12], f[7, 10, 11], 
  f[7, 11, 10], f[7, 12, 10], f[7, 13, 9], f[7, 15, 8], f[8, 3, 14], 
  f[8, 4, 14], f[8, 5, 13], f[8, 5, 14], f[8, 6, 13], f[8, 7, 12], 
  f[8, 7, 13], f[8, 8, 12], f[8, 9, 11], f[8, 10, 11], f[8, 11, 10], 
  f[8, 13, 9], f[9, 3, 14], f[9, 4, 14], f[9, 5, 13], f[9, 5, 14], 
  f[9, 6, 13], f[9, 7, 12], f[9, 8, 12], f[9, 9, 11], f[9, 11, 10], 
  f[10, 3, 14], f[10, 4, 14], f[10, 5, 13], f[10, 6, 13], 
  f[10, 7, 12], f[10, 9, 11], f[11, 3, 14], f[11, 4, 14], 
  f[11, 5, 13], f[11, 7, 12], f[12, 3, 14], f[12, 5, 13], 
  f[13, 3, 14]}}

which makes it seemingly difficult to express r as linear form in f[i1,i2,i3] because the first step would be to determine the needed variables f[i1,i2,i3] with respect to a,b,c,d,z.
POSTED BY: Udo Krause

Even this is miraculous, 

In[56]:= Length /@ %54
Out[56]= {5, 16, 34, 59, 88, 118, 146, 170, 188, 198, 198}

at least for big z the number of f[i1,i2,i3] terms stabilizes.
POSTED BY: Udo Krause

<deleted> The previous post appeared twice </deleted>
POSTED BY: Udo Krause
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