Av(1234, 1243, 1324, 2413)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x^{2}+2 x -1\right)}{\left(x -1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 214, 686, 2186, 6946, 22042, 69906, 221650, 702706, 2227714, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = -2 a \! \left(n \right)-2 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = -2 a \! \left(n \right)-2 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{\left(\left(4144 \,\mathrm{I} \sqrt{3}-616 \,\mathrm{I} \sqrt{37}\, \sqrt{3}+1848 \sqrt{37}+4144\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}-58016+\left(-137 \,\mathrm{I} \sqrt{37}\, \sqrt{3}+185 \,\mathrm{I} \sqrt{3}-411 \sqrt{37}-185\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-392 \,\mathrm{I} \sqrt{3}-392\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}}{4704}-\frac{1}{3}+\frac{\left(\left(3 \,\mathrm{I} \sqrt{37}-67 \,\mathrm{I}\right) \sqrt{3}+9 \sqrt{37}+67\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{4704}\right)^{-n}}{174048}+\frac{\left(\left(-616 \,\mathrm{I} \sqrt{37}\, \sqrt{3}-4144 \,\mathrm{I} \sqrt{3}-1848 \sqrt{37}+4144\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}-58016+\left(-137 \,\mathrm{I} \sqrt{37}\, \sqrt{3}-185 \,\mathrm{I} \sqrt{3}+411 \sqrt{37}-185\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(392 \,\mathrm{I} \sqrt{3}-392\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}}{4704}-\frac{1}{3}+\frac{\left(\left(3 \,\mathrm{I} \sqrt{37}+67 \,\mathrm{I}\right) \sqrt{3}-9 \sqrt{37}+67\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{4704}\right)^{-n}}{174048}+2+\frac{\left(\left(1232 \,\mathrm{I} \sqrt{37}\, \sqrt{3}-8288\right) \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}+274 \,\mathrm{I} \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{37}\, \sqrt{3}+370 \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}-58016\right) \left(\frac{\left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}-\frac{67 \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{2352}-\frac{\mathrm{I} \left(-134+6 \,\mathrm{I} \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{37}\, \sqrt{3}}{784}\right)^{-n}}{174048}\)
This specification was found using the strategy pack "Point Placements" and has 49 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 49 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{10}\! \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_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \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_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{29}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{10}\! \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_{35}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \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_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
\end{align*}\)