Av(1234, 1243, 1324, 3214)
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Generating Function
\(\displaystyle \frac{x^{6}-x^{3}-2 x^{2}-2 x +1}{x^{6}-x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 66, 218, 719, 2373, 7832, 25849, 85313, 281570, 929304, 3067109, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-x^{2}-3 x +1\right) F \! \left(x \right)-x^{6}+x^{3}+2 x^{2}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 218\)
\(\displaystyle a \! \left(n \right) = a \! \left(n +4\right)+3 a \! \left(5+n \right)-a \! \left(n +6\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{120207 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{890653}+\\\frac{32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{890653}-\\\frac{13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{890653}-\\\frac{12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{890653}-\\\frac{41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +5}}{890653}+\\\frac{\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{890653}\\+\\\frac{\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{890653}\\+\\\frac{\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{890653}\\+\\\frac{\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +4}}{890653}\\+\\\frac{\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{890653}\\+\\\frac{\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{890653}\\+\\\frac{\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +3}}{890653}\\+\\\frac{\left(\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}+\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-25512 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{890653}\\+\\\frac{\left(\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +2}}{890653}\\+\\\frac{\left(\left(\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +1}}{890653}\\+\frac{111289 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{890653}\\+\\\frac{\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{5}-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12053\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{890653}\\+\\\frac{\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{4}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}-24809\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{890653}\\+\\\frac{\left(\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{3}+\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-25512 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}+\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-25512 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-25512 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-24809\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{890653}\\+\\\frac{\left(\left(\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}+\left(\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{2}+\left(-41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+38268 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-27672\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-27672 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{2}+\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-27672 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{2}+\left(12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-27672 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{2}+\left(-13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)-32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-24809\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{890653}\\+\\\frac{\left(\left(\left(\left(\left(41114 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)-12756\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+\left(13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)-79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =3\right)+\left(\left(\left(-12756 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+13836\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+\left(13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)-79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\left(13836 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)+32751\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)+32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)-79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =4\right)+\left(32751 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)-79093\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =1\right)-79093 \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =5\right)-24809\right) \mathit{RootOf}\left(Z^{6}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n}}{890653} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 71 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{14}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \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_{53}\! \left(x \right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{40}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \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_{63}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)+F_{57}\! \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_{61}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{46}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{52}\! \left(x \right)\\ \end{align*}\)