Av(1243, 1342, 1432, 2143, 4213)
Generating Function
\(\displaystyle -\frac{\left(x^{2}+x -1\right) \left(x^{4}-2 x^{3}+x^{2}-2 x +1\right)}{\left(x -1\right) \left(2 x^{3}-x^{2}+3 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 165, 477, 1381, 3997, 11565, 33461, 96813, 280109, 810437, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+\left(x^{2}+x -1\right) \left(x^{4}-2 x^{3}+x^{2}-2 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+1, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+1, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(-6545 \sqrt{211}-272612 \,\mathrm{I}\right) \sqrt{3}-19635 \,\mathrm{I} \sqrt{211}-272612\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}-853706+\left(\left(3464 \sqrt{211}+69419 \,\mathrm{I}\right) \sqrt{3}-10392 \,\mathrm{I} \sqrt{211}-69419\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}-\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{8780976}\\+\\\frac{\left(\left(\left(3464 \sqrt{211}-69419 \,\mathrm{I}\right) \sqrt{3}+10392 \,\mathrm{I} \sqrt{211}-69419\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}-853706+\left(\left(-6545 \sqrt{211}+272612 \,\mathrm{I}\right) \sqrt{3}+19635 \,\mathrm{I} \sqrt{211}-272612\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}+\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{8780976}\\-\frac{1}{3}+\\\frac{\left(\left(-6928 \sqrt{211}\, \sqrt{3}+138838\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}+13090 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{211}\, \sqrt{3}+545224 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}-853706\right) \left(\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}+\frac{14 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{867}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}}{578}\right)^{-n}}{8780976} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 95 rules.
Found on January 18, 2022.Finding the specification took 1 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{81}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= 3 F_{35}\! \left(x \right)+F_{89}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
\end{align*}\)