Av(13425, 13452, 13542, 14325, 14352, 31425, 31452, 31542, 34125, 34152, 34512, 35142, 35412, 41325, 41352, 43125, 43152, 43512)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x^{4}-2 x^{3}-4 x^{2}+5 x -1\right)}{\left(x^{2}+x -1\right) \left(9 x^{3}-14 x^{2}+7 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 102, 433, 1820, 7598, 31606, 131255, 544691, 2259734, 9373780, 38882437, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(9 x^{3}-14 x^{2}+7 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{4}-2 x^{3}-4 x^{2}+5 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a{\left(n + 5 \right)} = - 9 a{\left(n \right)} + 5 a{\left(n + 1 \right)} + 16 a{\left(n + 2 \right)} - 20 a{\left(n + 3 \right)} + 8 a{\left(n + 4 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a{\left(n + 5 \right)} = - 9 a{\left(n \right)} + 5 a{\left(n + 1 \right)} + 16 a{\left(n + 2 \right)} - 20 a{\left(n + 3 \right)} + 8 a{\left(n + 4 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(4805 \left(\left(i-\frac{45 \sqrt{31}}{961}\right) \sqrt{3}-\frac{135 i \sqrt{31}}{961}+1\right) 2^{\frac{1}{3}} \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+607600-49910 \,2^{\frac{2}{3}} \left(\left(i+\frac{72 \sqrt{31}}{713}\right) \sqrt{3}-\frac{216 i \sqrt{31}}{713}-1\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{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}}{1640520}\\+\\\frac{\left(49910 \,2^{\frac{2}{3}} \left(\left(i-\frac{72 \sqrt{31}}{713}\right) \sqrt{3}-\frac{216 i \sqrt{31}}{713}+1\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+607600-4805 \left(\left(i+\frac{45 \sqrt{31}}{961}\right) \sqrt{3}-\frac{135 i \sqrt{31}}{961}-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}}{1640520}\\+\\\frac{\left(\left(10080 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}-99820 \,2^{\frac{2}{3}}\right) \left(263+27 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+607600+\left(450 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-9610 \,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}}{1640520}\\+\frac{\left(164052 \sqrt{5}-273420\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{1640520}-\frac{\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \left(\sqrt{5}+\frac{5}{3}\right)}{10} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 68 rules.
Finding the specification took 108 seconds.
Copy 68 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_{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_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \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_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
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_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{23}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{23}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{16}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 0\\
F_{63}\! \left(x \right) &= F_{16}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{56}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 0\\
\end{align*}\)