Av(1234, 1324, 1342, 1432, 4213)
View Raw Data
Generating Function
\(\displaystyle \frac{3 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}-2 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 129, 304, 699, 1570, 3469, 7591, 16487, 35600, 76549, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}+x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)-3 x^{7}-2 x^{6}+8 x^{5}+5 x^{4}+2 x^{3}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 129\)
\(\displaystyle a \! \left(7\right) = 304\)
\(\displaystyle a \! \left(n +2\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)+a \! \left(n +4\right)+2 a \! \left(n +5\right)-a \! \left(n +6\right)+9, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{13973 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{1325}+\frac{67598 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{3975}-\frac{43891 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{7950}-\frac{4347 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{530}-\frac{12899 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{1325}-\frac{45544 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{3975}+\frac{36427 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+2 Z^{6}-3 Z^{5}-Z^{4}-Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{3975}+\left(\left\{\begin{array}{cc}\frac{3}{2} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)

This specification was found using the strategy pack "Point Placements" and has 114 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_{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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \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_{6}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \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_{25}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{99}\! \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_{51}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{57}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \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_{15}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{78}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \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) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= x^{2}\\ F_{83}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{86}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{69}\! \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_{102}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \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_{110}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{35}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{111}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{112}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{32}\! \left(x \right)\\ \end{align*}\)