Av(1234, 1243, 3214)
Generating Function
\(\displaystyle \frac{2 x^{5}-x^{4}-3 x^{3}-2 x^{2}-2 x +1}{2 x^{5}-2 x^{3}-x^{2}-3 x +1}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 250, 861, 2967, 10220, 35203, 121263, 417710, 1438865, 4956391, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-2 x^{3}-x^{2}-3 x +1\right) F \! \left(x \right)-2 x^{5}+x^{4}+3 x^{3}+2 x^{2}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(n \right) = a \! \left(n +2\right)+\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(4+n \right)}{2}-\frac{a \! \left(n +5\right)}{2}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(n \right) = a \! \left(n +2\right)+\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(4+n \right)}{2}-\frac{a \! \left(n +5\right)}{2}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{7337 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{41583}\\+\\\frac{5657 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{41583}\\-\\\frac{296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{41583}\\-\\\frac{3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{41583}\\+\\\frac{\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+5517\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{41583}\\+\\\frac{\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{41583}\\+\\\frac{\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{41583}\\+\\\frac{\left(\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+5221\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{41583}\\+\\\frac{\left(\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{41583}\\+\\\frac{\left(\left(\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+5221\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{41583}\\+\\\frac{3650 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{41583}\\+\\\frac{\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-1820 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-1810\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{41583}\\+\\\frac{\left(\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-1958\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{41583}\\+\\\frac{\left(\left(\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}+\left(3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-592 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}+\left(-296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+59\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{41583}\\+\\\frac{\left(\left(\left(\left(-3640 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+296\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-5221\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\left(296 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+2017\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-5221\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(2017 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-5221\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-5221 \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+59\right) \mathit{RootOf}\left(2 Z^{5}-2 Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{41583} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 80 rules.
Found on January 18, 2022.Finding the specification took 10 seconds.
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Copy 80 equations to clipboard:
\(\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_{19}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{29}\! \left(x \right) &= 0\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \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_{39}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 0\\
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_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\
\end{align*}\)