Av(1234, 1324, 1432, 2413, 3241)
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Generating Function
\(\displaystyle \frac{3 x^{8}+8 x^{7}+3 x^{6}-6 x^{5}-4 x^{4}-2 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 51, 121, 280, 645, 1465, 3286, 7327, 16282, 36067, 79693, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)-3 x^{8}-8 x^{7}-3 x^{6}+6 x^{5}+4 x^{4}+2 x^{3}-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 121\)
\(\displaystyle a \! \left(7\right) = 280\)
\(\displaystyle a \! \left(8\right) = 645\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{4}-a \! \left(n +1\right)+\frac{a \! \left(n +5\right)}{4}+\frac{a \! \left(n +6\right)}{2}-\frac{a \! \left(n +7\right)}{4}-\frac{3}{4}, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left\{\begin{array}{cc}4920 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{4}\\+\\4920 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{4}\\+\\4920 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{4}\\+\\4920 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{4}\\+\\4920 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{4}\\+\\37593 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}\\+\\37593 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}\\+\\37593 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{3}\\+\\37593 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{3}\\+\\37593 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{3}\\+\\55161 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}\\+\\55161 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}\\+\\55161 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}\\+\\55161 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{2}\\+\\55161 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{2}\\+\\46494 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)\\+\\46494 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)\\+\\46494 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)\\+\\46494 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)\\+\\46494 \\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)\\+243362 & n =0 \\ 4920 \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{3}\right. \\+\\2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)\right. \\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{3}\right. \\+\\2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)\right. \\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)\right. \\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}\right. \\+\\2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{3}\right. \\+\\2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)\right. \\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)\right. \\ \left. +1\right)\\ \\\left(\frac{5509 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +1} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{984}\right. \\+\\\frac{169 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{20}\\+\\\frac{15107 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{7611 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{5509 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +1} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{984}\\+\\\frac{169 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{20}\\+\\\frac{15107 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{7611 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{5509 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +1} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{984}\\+\\\frac{169 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{20}\\+\\\frac{15107 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{7611 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{5509 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +1} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{984}\\+\\\frac{169 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{20}\\+\\\frac{15107 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{7611 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{1640}\\+\\\frac{5509 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +1} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{984}\\+\\\frac{169 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{20}\\+\\\frac{15107 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{1640}\\+\\\frac{7611 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +4} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{1640}\\+\\\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)\\+\\ \left. \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}+\frac{51569 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right) \left(\left(\sqrt{5}-1\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-\frac{6}{7}+\left(-\sqrt{5}-1\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}\right)}{9840}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)\right)\\ \\\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}\right. \\+\\2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}\\ \left. +1\right) & \text{otherwise} \end{array}\right.\right)}{51569}\)
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This specification was found using the strategy pack "Point Placements" and has 190 rules.

Found on January 18, 2022.

Finding the specification took 8 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_{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_{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_{17}\! \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_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ 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_{35}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{40}\! \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_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{49}\! \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_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{65}\! \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_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{70}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{123}\! \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_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{87}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{91}\! \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_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{102}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{103}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{104}\! \left(x \right)+F_{95}\! \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_{101}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{113}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{132}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{121}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{184}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{180}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{150}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{123}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{171}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{163}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{160}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{160}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{171}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{121}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{175}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{154}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{154}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{176}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{187}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{189}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{185}\! \left(x \right)\\ \end{align*}\)