Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432, 21354, 21453, 21543, 23154, 24153, 25143, 31254, 32154)
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Counting Sequence
1, 1, 2, 6, 24, 100, 426, 1875, 8482, 39143, 183476, 871216, 4182214, 20263013, 98958258, ...

This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 52 rules.

Found on January 20, 2022.

Finding the specification took 91 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{35}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{31}\! \left(x , y\right)+F_{33}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= y x\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x , y\right) &= -\frac{-y F_{27}\! \left(x , y\right)+F_{27}\! \left(x , 1\right)}{-1+y}\\ F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= -\frac{-y F_{25}\! \left(x , y\right)+F_{25}\! \left(x , 1\right)}{-1+y}\\ F_{33}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{32}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= -\frac{-y F_{37}\! \left(x , y\right)+F_{37}\! \left(x , 1\right)}{-1+y}\\ F_{37}\! \left(x , y\right) &= -\frac{-y F_{22}\! \left(x , y\right)+F_{22}\! \left(x , 1\right)}{-1+y}\\ F_{38}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\ F_{39}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\ F_{40}\! \left(x \right) &= F_{3}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x , 1\right)\\ F_{42}\! \left(x , y\right) &= -\frac{-y F_{22}\! \left(x , y\right)+F_{22}\! \left(x , 1\right)}{-1+y}\\ F_{43}\! \left(x \right) &= F_{3}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{3}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x , 1\right)\\ F_{47}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{48}\! \left(x \right) &= F_{3}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\ F_{50}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{51}\! \left(x \right) &= F_{3}\! \left(x \right) F_{49}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob" and has 53 rules.

Found on January 20, 2022.

Finding the specification took 561 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{32}\! \left(x , y\right)+F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= y x\\ F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{25}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= -\frac{-y F_{27}\! \left(x , y\right)+F_{27}\! \left(x , 1\right)}{-1+y}\\ F_{27}\! \left(x , y\right) &= -\frac{-y F_{28}\! \left(x , y\right)+F_{28}\! \left(x , 1\right)}{-1+y}\\ F_{28}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{34}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x , y\right) &= -\frac{-y F_{38}\! \left(x , y\right)+F_{38}\! \left(x , 1\right)}{-1+y}\\ F_{38}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\ F_{39}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\ F_{40}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x , 1\right)\\ F_{43}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x , 1\right)\\ F_{48}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x , 1\right)\\ F_{51}\! \left(x , y\right) &= -\frac{-y F_{27}\! \left(x , y\right)+F_{27}\! \left(x , 1\right)}{-1+y}\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ \end{align*}\)