Av(1234, 1243, 1342, 1432)
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Generating Function
\(\displaystyle -\frac{3 x -1}{2 x^{2}-4 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}-4 x +1\right) F \! \left(x \right)+3 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = -2 a \! \left(n \right)+4 a \! \left(n +1\right), \quad n \geq 2\)
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Explicit Closed Form
\(\displaystyle -\frac{\left(2+\sqrt{2}\right) \left(2 \left(1-\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-\left(1+\frac{\sqrt{2}}{2}\right)^{-n}-3 \left(1-\frac{\sqrt{2}}{2}\right)^{-n}\right)}{4}\)
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This specification was found using the strategy pack "Point Placements" and has 133 rules.

Found on January 18, 2022.

Finding the specification took 5 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_{17}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ 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_{31}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{37}\! \left(x \right) &= 0\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 0\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{53}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{19}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{19}\! \left(x \right)+F_{75}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{65}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{124}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{122}\! \left(x \right)+F_{19}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{120}\! \left(x \right)+F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{121}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{125}\! \left(x \right)\\ \end{align*}\)