Av(1234, 1243, 2314, 4123, 4213)
Generating Function
\(\displaystyle -\frac{2 x^{5}-2 x^{4}-4 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 146, 362, 853, 1935, 4266, 9204, 19533, 40931, 84932, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{5}-2 x^{4}-4 x^{3}+9 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}-a \! \left(n +1\right)+3 a \! \left(n +2\right)-2 a \! \left(n \right)+\frac{3 n}{2}+3, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}-a \! \left(n +1\right)+3 a \! \left(n +2\right)-2 a \! \left(n \right)+\frac{3 n}{2}+3, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{\left(36 \sqrt{5}-80\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(-36 \sqrt{5}-80\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{n^{2}}{2}+\frac{7 n}{2}+6 \,2^{n}+11\)
This specification was found using the strategy pack "Point Placements" and has 112 rules.
Found on January 18, 2022.Finding the specification took 2 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_{24}\! \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_{18}\! \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_{10}\! \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_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \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_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \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_{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_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \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_{47}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{56}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{82}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \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_{72}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{19}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{106}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{107}\! \left(x \right)\\
\end{align*}\)