Av(1243, 1342, 2314, 3124, 4123)
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Generating Function
\(\displaystyle \frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{4}+3 x^{2}-3 x +1\right)}{2 x \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 170, 522, 1666, 5503, 18665, 64565, 226675, 805079, 2886349, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x \right)^{2}+\left(x^{4}+3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x^{4}+3 x^{2}-3 x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 170\)
\(\displaystyle a \! \left(7\right) = 522\)
\(\displaystyle a \! \left(8\right) = 1666\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(-9+5 n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(41+13 n \right) a \! \left(n +2\right)}{7+n}+\frac{9 \left(11+3 n \right) a \! \left(n +3\right)}{7+n}-\frac{2 \left(51+11 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}+\frac{4 n}{7+n}, \quad n \geq 9\)
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This specification was found using the strategy pack "Row And Col Placements Req Corrob Expand Verified" and has 31 rules.

Found on January 21, 2022.

Finding the specification took 31 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_{16}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= \frac{y F_{12}\! \left(x , y\right)-F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25} \left(x \right)^{2} F_{28}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{29}\! \left(x \right) F_{3}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row And Col Placements Expand Verified" and has 79 rules.

Found on January 21, 2022.

Finding the specification took 30 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_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x , 1\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) &= F_{1}\! \left(x \right)+F_{38}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= y x\\ F_{40}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{30}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 0\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= -F_{59}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{11}\! \left(x \right) F_{47}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{63}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= -F_{71}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= \frac{F_{66}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= -F_{49}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= \frac{F_{70}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{70}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{30}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{59}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{10}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{47}\! \left(x \right) F_{49}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Expand Verified" and has 71 rules.

Found on January 21, 2022.

Finding the specification took 33 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_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x , 1\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) &= F_{1}\! \left(x \right)+F_{38}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= y x\\ F_{40}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{30}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{49}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{55}\! \left(x \right) &= -F_{56}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{47}\! \left(x \right) F_{50}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{60}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= -F_{68}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= -F_{49}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{67}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{30}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{50} \left(x \right)^{3} F_{56}\! \left(x \right)\\ \end{align*}\)