Av(1342, 2143, 4123)
Generating Function
\(\displaystyle -\frac{3 x^{4}-5 x^{3}+8 x^{2}-5 x +1}{\left(x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 21, 72, 237, 761, 2415, 7626, 24034, 75689, 238298, 750179, 2361533, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+3 x^{4}-5 x^{3}+8 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2 n, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2 n, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{\left(9 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{1}{6}}}{3}-\frac{31 \,3^{\frac{2}{3}}}{9}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-248+6 \left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) 2^{\frac{2}{3}} \sqrt{31}\, \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(-\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{248}+\frac{\left(-9 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{1}{6}}}{3}+\frac{31 \,3^{\frac{2}{3}}}{9}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-248-6 \,2^{\frac{2}{3}} \left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}\, \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{248}+\frac{\left(-18 \left(3^{\frac{1}{6}} \sqrt{31}-\frac{31 \,3^{\frac{2}{3}}}{9}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+4 \,2^{\frac{2}{3}} 3^{\frac{5}{6}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{31}-248\right) \left(-\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+1+\frac{\left(\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}\right)^{-n}}{248}-2 n +4\)
This specification was found using the strategy pack "Point Placements" and has 108 rules.
Found on January 18, 2022.Finding the specification took 3 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_{41}\! \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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \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_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \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_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 0\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{60}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{60}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{71}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 2 F_{47}\! \left(x \right)+F_{71}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 2 F_{47}\! \left(x \right)+F_{71}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{98}\! \left(x \right) &= 2 F_{47}\! \left(x \right)+F_{71}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 2 F_{47}\! \left(x \right)+F_{71}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{104}\! \left(x \right)\\
\end{align*}\)