Av(13425, 13452, 14235, 14253, 14325, 14352, 14523, 14532, 31425, 31452, 34125, 34152, 34512, 41235, 41253, 41325, 41352, 41523, 41532, 43125, 43152, 43512, 45123, 45132, 45312)
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Generating Function
\(\displaystyle \frac{5 x^{6}-17 x^{5}+29 x^{4}-35 x^{3}+24 x^{2}-8 x +1}{\left(5 x^{2}-5 x +1\right) \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 358, 1312, 4756, 17189, 62110, 224518, 811912, 2936791, 10624154, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(5 x^{2}-5 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-5 x^{6}+17 x^{5}-29 x^{4}+35 x^{3}-24 x^{2}+8 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 358\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{n^{3} - 25 n + 18}{6} - 5 a{\left(n \right)} + 5 a{\left(n + 1 \right)}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-132 \sqrt{5}+300\right) \left(\frac{1}{2}-\frac{\sqrt{5}}{10}\right)^{-n}}{30}+\frac{\left(132 \sqrt{5}+300\right) \left(\frac{1}{2}+\frac{\sqrt{5}}{10}\right)^{-n}}{30}-\frac{n^{3}}{6}-\frac{3 n^{2}}{2}\\-\frac{16 n}{3}-20 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 80 rules.

Finding the specification took 81 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{40}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{40}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{40}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{16}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{62}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{16}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{76}\! \left(x \right)\\ \end{align*}\)