Av(12453, 12534, 12543, 13452, 13524, 13542, 14352, 14523, 14532, 15342, 15423, 15432, 23451, 23541, 24351, 24531, 25341, 25431, 34251, 35241)
Generating Function
\(\displaystyle -\frac{3 x^{9}-5 x^{8}-13 x^{7}+33 x^{6}-5 x^{5}-40 x^{4}+52 x^{3}-31 x^{2}+9 x -1}{\left(x -1\right) \left(3 x^{7}-10 x^{6}+5 x^{5}+19 x^{4}-32 x^{3}+23 x^{2}-8 x +1\right) \left(x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 405, 1591, 6143, 23542, 89980, 343613, 1311767, 5007047, 19110625, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(3 x^{7}-10 x^{6}+5 x^{5}+19 x^{4}-32 x^{3}+23 x^{2}-8 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+3 x^{9}-5 x^{8}-13 x^{7}+33 x^{6}-5 x^{5}-40 x^{4}+52 x^{3}-31 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 405\)
\(\displaystyle a(7) = 1591\)
\(\displaystyle a(8) = 6143\)
\(\displaystyle a(9) = 23542\)
\(\displaystyle a{\left(n + 9 \right)} = 3 a{\left(n \right)} - 7 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 34 a{\left(n + 3 \right)} - 18 a{\left(n + 4 \right)} - 28 a{\left(n + 5 \right)} + 47 a{\left(n + 6 \right)} - 30 a{\left(n + 7 \right)} + 9 a{\left(n + 8 \right)} - 2, \quad n \geq 10\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 405\)
\(\displaystyle a(7) = 1591\)
\(\displaystyle a(8) = 6143\)
\(\displaystyle a(9) = 23542\)
\(\displaystyle a{\left(n + 9 \right)} = 3 a{\left(n \right)} - 7 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 34 a{\left(n + 3 \right)} - 18 a{\left(n + 4 \right)} - 28 a{\left(n + 5 \right)} + 47 a{\left(n + 6 \right)} - 30 a{\left(n + 7 \right)} + 9 a{\left(n + 8 \right)} - 2, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \frac{1444338063 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +8}\right)}{26300105}-\frac{3866107359 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +7}\right)}{26300105}-\frac{3001857264 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +6}\right)}{26300105}+\frac{18224534664 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{26300105}-\frac{13152882189 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{26300105}-\frac{13287215013 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{26300105}+\frac{27416251164 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{26300105}-\frac{19283356594 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{26300105}+\frac{6306320026 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{26300105}-\frac{747425288 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{10}-10 Z^{9}-Z^{8}+42 Z^{7}-52 Z^{6}-10 Z^{5}+75 Z^{4}-77 Z^{3}+39 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{26300105}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 101 rules.
Finding the specification took 123 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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{78}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{16}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
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_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{69}\! \left(x \right)+F_{7}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \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_{19}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \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_{44}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{28}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{16}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{61}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{16}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{51}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{68}\! \left(x \right) &= 0\\
F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{16}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 0\\
F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{7}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{16}\! \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_{87}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{16}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{16}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{7}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{16}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{100}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{51}\! \left(x \right)+F_{97}\! \left(x \right)\\
\end{align*}\)