Av(12345, 12354, 12453, 13245, 13254, 13452, 14235, 14253, 14352, 15234, 15243, 15342, 23145, 23154, 24135, 24153, 25134, 25143, 34125, 35124)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right)}{14 x^{3}-16 x^{2}+7 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 400, 1536, 5752, 21288, 78488, 289336, 1067576, 3942488, 14566904, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(14 x^{3}-16 x^{2}+7 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = 14 a{\left(n \right)} - 16 a{\left(n + 1 \right)} + 7 a{\left(n + 2 \right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = 14 a{\left(n \right)} - 16 a{\left(n + 1 \right)} + 7 a{\left(n + 2 \right)}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-1073 \,2^{\frac{2}{3}} \left(\left(i-\frac{224 \sqrt{29}}{1073}\right) \sqrt{3}-\frac{672 i \sqrt{29}}{1073}+1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}+335008+12673 \,2^{\frac{1}{3}} \left(\left(i+\frac{35 \sqrt{29}}{667}\right) \sqrt{3}-\frac{105 i \sqrt{29}}{667}-1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{157 \,2^{\frac{2}{3}} \left(\left(i-\frac{21 \sqrt{29}}{157}\right) \sqrt{3}-\frac{63 i \sqrt{29}}{157}+1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{60648}-\frac{i \sqrt{3}\, \left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{8}{21}\right)^{-n}}{1758792}\\+\\\frac{\left(-12673 \,2^{\frac{1}{3}} \left(\left(i-\frac{35 \sqrt{29}}{667}\right) \sqrt{3}-\frac{105 i \sqrt{29}}{667}+1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}+335008+1073 \left(\left(i+\frac{224 \sqrt{29}}{1073}\right) \sqrt{3}-\frac{672 i \sqrt{29}}{1073}-1\right) 2^{\frac{2}{3}} \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{157 \,2^{\frac{2}{3}} \left(\left(i+\frac{21 \sqrt{29}}{157}\right) \sqrt{3}-\frac{63 i \sqrt{29}}{157}-1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{60648}+\frac{i \sqrt{3}\, \left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{8}{21}\right)^{-n}}{1758792}\\-\\\frac{5 \left(2^{\frac{1}{3}} \left(\sqrt{29}\, \sqrt{3}-\frac{667}{35}\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{8816}{35}+\left(\frac{32 \,2^{\frac{2}{3}} \sqrt{29}\, \sqrt{3}}{95}-\frac{1073 \,2^{\frac{2}{3}}}{665}\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{2}{3}} \left(21 \sqrt{29}\, \sqrt{3}-157\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{30324}-\frac{\left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{42}+\frac{8}{21}\right)^{-n}}{6612} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 114 rules.
Finding the specification took 115 seconds.
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Copy 114 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_{18}\! \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_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{18}\! \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_{20}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{37}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{18}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{18}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{18}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{61}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{18}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{18}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{18}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 0\\
F_{70}\! \left(x \right) &= F_{18}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{67}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 0\\
F_{75}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{18}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{69}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{80}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{18}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{18}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{88}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{18}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{21}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{113}\! \left(x \right)+F_{21}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{111}\! \left(x \right)+F_{21}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{111}\! \left(x \right) &= 0\\
F_{112}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{113}\! \left(x \right) &= 0\\
\end{align*}\)