Av(13425, 13452, 13542, 14235, 14325, 14352, 31425, 31452, 31542, 34125, 34152, 34512, 35142, 35412, 41235, 41325, 41352, 43125, 43152, 43512)
Generating Function
\(\displaystyle \frac{x^{6}-4 x^{5}-x^{4}+5 x^{3}-10 x^{2}+6 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(3 x^{3}-5 x^{2}+5 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 407, 1624, 6415, 25227, 99017, 388335, 1522500, 5968249, 23394356, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}+x -1\right) \left(3 x^{3}-5 x^{2}+5 x -1\right) F \! \left(x \right)-x^{6}+4 x^{5}+x^{4}-5 x^{3}+10 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 407\)
\(\displaystyle a{\left(n + 5 \right)} = - 3 a{\left(n \right)} + 2 a{\left(n + 1 \right)} + 3 a{\left(n + 2 \right)} - 9 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)} + 4, \quad n \geq 7\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 407\)
\(\displaystyle a{\left(n + 5 \right)} = - 3 a{\left(n \right)} + 2 a{\left(n + 1 \right)} + 3 a{\left(n + 2 \right)} - 9 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)} + 4, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-92728 i-8472 \sqrt{67}\right) \sqrt{3}+25416 i \sqrt{67}+92728\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}-1050560+\left(\left(-17219 i-159 \sqrt{67}\right) \sqrt{3}-477 i \sqrt{67}-17219\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(91 i-9 \sqrt{67}\right) \sqrt{3}-27 i \sqrt{67}+91\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}}{7200}-\frac{i \sqrt{3}\, \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{\left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{5981760}\\+\\\frac{\left(\left(\left(17219 i-159 \sqrt{67}\right) \sqrt{3}+477 i \sqrt{67}-17219\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}-1050560+\left(\left(92728 i-8472 \sqrt{67}\right) \sqrt{3}-25416 i \sqrt{67}+92728\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-91 i-9 \sqrt{67}\right) \sqrt{3}+27 i \sqrt{67}+91\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}}{7200}+\frac{i \sqrt{3}\, \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{\left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{5981760}\\+\\\frac{\left(\left(318 \sqrt{67}\, \sqrt{3}+34438\right) \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}+16944 \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{67}\, \sqrt{3}-185456 \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}-1050560\right) \left(-\frac{\left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}+\frac{5}{9}-\frac{91 \left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}}{3600}+\frac{\left(91+9 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{67}\, \sqrt{3}}{400}\right)^{-n}}{5981760}\\+\frac{\left(1099872 \sqrt{5}-2412000\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5981760}+2+\\\frac{\left(-1099872 \sqrt{5}-2412000\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5981760} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 77 rules.
Finding the specification took 141 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_{21}\! \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_{19}\! \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_{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_{16}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{51}\! \left(x \right)+F_{54}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{16}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 0\\
F_{71}\! \left(x \right) &= F_{16}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{43}\! \left(x \right)+F_{64}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 0\\
\end{align*}\)