Av(12453, 12534, 12543, 13452, 13542, 21453, 21534, 21543, 23451, 23541, 31452, 31524, 31542, 32451, 32541, 41523, 41532, 42531)
Generating Function
\(\displaystyle \frac{5 x^{5}-16 x^{4}+27 x^{3}-22 x^{2}+8 x -1}{\left(9 x^{3}-14 x^{2}+7 x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 102, 431, 1803, 7502, 31147, 129223, 536015, 2223299, 9221882, 38251114, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(9 x^{3}-14 x^{2}+7 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-5 x^{5}+16 x^{4}-27 x^{3}+22 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a{\left(n + 3 \right)} = - n + 9 a{\left(n \right)} - 14 a{\left(n + 1 \right)} + 7 a{\left(n + 2 \right)} + 2, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a{\left(n + 3 \right)} = - n + 9 a{\left(n \right)} - 14 a{\left(n + 1 \right)} + 7 a{\left(n + 2 \right)} + 2, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-8122 \left(\left(-\frac{432 \sqrt{31}}{4061}+i\right) \sqrt{3}-\frac{1296 i \sqrt{31}}{4061}+1\right) 2^{\frac{1}{3}} \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-30380-2387 \left(\left(\frac{81 \sqrt{31}}{341}+i\right) \sqrt{3}-\frac{243 i \sqrt{31}}{341}-1\right) 2^{\frac{2}{3}} \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{263 \left(\left(i-\frac{27 \sqrt{31}}{263}\right) \sqrt{3}-\frac{81 i \sqrt{31}}{263}+1\right) 2^{\frac{1}{3}} \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{10584}-\frac{i \sqrt{3}\, \left(1052+108 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{108}+\frac{\left(1052+108 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{108}+\frac{14}{27}\right)^{-n}}{164052}\\+\\\frac{\left(2387 \left(\left(-\frac{81 \sqrt{31}}{341}+i\right) \sqrt{3}-\frac{243 i \sqrt{31}}{341}+1\right) 2^{\frac{2}{3}} \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-30380+8122 \left(\left(\frac{432 \sqrt{31}}{4061}+i\right) \sqrt{3}-\frac{1296 i \sqrt{31}}{4061}-1\right) 2^{\frac{1}{3}} \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{263 \,2^{\frac{1}{3}} \left(\left(i+\frac{27 \sqrt{31}}{263}\right) \sqrt{3}-\frac{81 i \sqrt{31}}{263}-1\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{10584}+\frac{i \sqrt{3}\, \left(1052+108 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{108}+\frac{\left(1052+108 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{108}+\frac{14}{27}\right)^{-n}}{164052}\\+\\\frac{\left(\left(1134 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{31}-4774 \,2^{\frac{2}{3}}\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-30380+\left(-1728 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{31}+16244 \,2^{\frac{1}{3}}\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(27 \sqrt{31}\, \sqrt{3}-263\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{5292}-\frac{\left(1052+108 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{54}+\frac{14}{27}\right)^{-n}}{164052}\\+n +1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 119 rules.
Finding the specification took 131 seconds.
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Copy 119 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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{28}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \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_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{18}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)+F_{7}\! \left(x \right)+F_{83}\! \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_{51}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{18}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{60}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{18}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{18}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{61}\! \left(x \right)\\
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_{66}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{18}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{61}\! \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_{78}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{60}\! \left(x \right)+F_{76}\! \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_{82}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{59}\! \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_{43}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{7}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{18}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 0\\
F_{91}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{18}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{55}\! \left(x \right)+F_{7}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{111}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{112}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{115}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{74}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{49}\! \left(x \right)+F_{7}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{11}\! \left(x \right)\\
\end{align*}\)