Av(1234, 1324, 1342, 2413, 3124)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{4}}{8 x^{5}-21 x^{4}+27 x^{3}-19 x^{2}+7 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 173, 512, 1512, 4461, 13154, 38775, 114290, 336878, 993015, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(8 x^{5}-21 x^{4}+27 x^{3}-19 x^{2}+7 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(n +5\right) = 8 a \! \left(n \right)-21 a \! \left(n +1\right)+27 a \! \left(n +2\right)-19 a \! \left(n +3\right)+7 a \! \left(4+n \right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(n +5\right) = 8 a \! \left(n \right)-21 a \! \left(n +1\right)+27 a \! \left(n +2\right)-19 a \! \left(n +3\right)+7 a \! \left(4+n \right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{17920 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{4757}\\+\\\frac{31095 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{4757}\\-\\\frac{28171 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{4757}\\+\\\frac{11896 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +4}}{4757}\\+\\\frac{\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{3}+124908 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}-160596 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+41332\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{19028}\\+\\\frac{\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}+124908 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-36216\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{19028}\\+\\\frac{\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{19028}\\+\\\frac{\left(\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{2}+\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}-137132 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+32088\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)-12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}+32088 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+76\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{19028}\\+\\\frac{\left(\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)-12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{19028}\\+\\\frac{\left(\left(\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)+\left(12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+4128 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-10760\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{19028}\\+\\\frac{4705 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{4757}\\+\\\frac{\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{4}+124908 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{3}-160596 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}+113012 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-22816\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{19028}\\+\\\frac{\left(\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{3}+\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}-137132 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+32088\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{2}+\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{3}-137132 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}+192684 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)-41256\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)-12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{3}+32088 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}-41256 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+6216\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{19028}\\+\\\frac{\left(\left(\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{2}+\left(\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}+\left(-47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{2}+149356 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-27960\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{2}-27960 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-10836\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)+\left(12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{2}+\left(12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{2}-27960 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-10836\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+4128 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{2}-10836 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+20148\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{19028}\\+\\\frac{\left(\left(\left(\left(47584 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)-12224\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)-4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+\left(-12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)-4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-4128 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)+10760\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)+\left(\left(-12224 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)-4128\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)-4128 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)+10760\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)+\left(-4128 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)+10760\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)+10760 \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)-8097\right) \mathit{RootOf}\left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{19028} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 54 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 54 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \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) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
\end{align*}\)