Av(1234, 1243, 1342, 2431, 3142)
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Generating Function
\(\displaystyle -\frac{3 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 155, 414, 1088, 2843, 7421, 19381, 50657, 132491, 346674, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n^{2}-6 n -1\right)}{6}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{\left(-27 \sqrt{5}+75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(27 \sqrt{5}+75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-\frac{3 n^{2}}{2}+\frac{10 n}{3}-4\)
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This specification was found using the strategy pack "Point Placements" and has 76 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{49}\! \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_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \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 \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{30}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\ 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_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{61}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)\\ \end{align*}\)