Av(1243, 1432, 2143, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{5}}{x^{9}-5 x^{8}+23 x^{7}-55 x^{6}+89 x^{5}-99 x^{4}+73 x^{3}-34 x^{2}+9 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 66, 211, 660, 2047, 6344, 19692, 61204, 190326, 591884, 1840478, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{9}-5 x^{8}+23 x^{7}-55 x^{6}+89 x^{5}-99 x^{4}+73 x^{3}-34 x^{2}+9 x -1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{5} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 211\)
\(\displaystyle a \! \left(7\right) = 660\)
\(\displaystyle a \! \left(8\right) = 2047\)
\(\displaystyle a \! \left(n +9\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+23 a \! \left(n +2\right)-55 a \! \left(n +3\right)+89 a \! \left(n +4\right)-99 a \! \left(n +5\right)+73 a \! \left(n +6\right)-34 a \! \left(n +7\right)+9 a \! \left(n +8\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +7}}{62602963177}+\frac{18327326180 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +7}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +6}}{62602963177}-\frac{85284313482 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +6}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +5}}{62602963177}+\frac{390071323255 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +5}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +4}}{62602963177}-\frac{865442870569 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +4}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +3}}{62602963177}+\frac{1296515030666 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +3}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +2}}{62602963177}-\frac{1302864019595 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +2}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +1}}{62602963177}+\frac{801310324590 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n -1}}{62602963177}+\frac{42013133937 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n -1}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n}}{62602963177}-\frac{274097999527 \mathit{RootOf} \left(Z^{9}-5 Z^{8}+23 Z^{7}-55 Z^{6}+89 Z^{5}-99 Z^{4}+73 Z^{3}-34 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n}}{62602963177}\)
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This specification was found using the strategy pack "Point Placements" and has 95 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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Copy 95 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{20}\! \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_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \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_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \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_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{45}\! \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_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \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_{2}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)+F_{94}\! \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_{75}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{33}\! \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_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{76}\! \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_{35}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{91}\! \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_{85}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ \end{align*}\)