Av(1243, 2413, 3241, 4132)
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Generating Function
\(\displaystyle -\frac{2 x^{6}-12 x^{5}+24 x^{4}-29 x^{3}+20 x^{2}-7 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}\)
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Counting Sequence
1, 1, 2, 6, 20, 63, 184, 509, 1364, 3597, 9421, 24621, 64332, 168166, 439815, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{6}-12 x^{5}+24 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 184\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n +1\right) \left(n^{2}-7 n -6\right)}{24}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{\left(12 \sqrt{5}-12\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{24}+\frac{\left(-12 \sqrt{5}-12\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{24}+\frac{n^{4}}{24}-\frac{5 n^{3}}{12}+\frac{23 n^{2}}{24}-\frac{31 n}{12}+2\)
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This specification was found using the strategy pack "Point Placements" and has 90 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{31}\! \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_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{24}\! \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_{48}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{12}\! \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_{10}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{12}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{12}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{12}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{80}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\ \end{align*}\)