Av(1243, 2341, 2413, 41352, 531642)
Generating Function
\(\displaystyle \frac{x^{7}-6 x^{6}+17 x^{5}-34 x^{4}+39 x^{3}-25 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(2 x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 245, 798, 2545, 8001, 24901, 76914, 236150, 721480, 2195132, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)-x^{7}+6 x^{6}-17 x^{5}+34 x^{4}-39 x^{3}+25 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 245\)
\(\displaystyle a \! \left(7\right) = 798\)
\(\displaystyle a \! \left(n +7\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+37 a \! \left(n +2\right)-62 a \! \left(n +3\right)+59 a \! \left(n +4\right)-32 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 245\)
\(\displaystyle a \! \left(7\right) = 798\)
\(\displaystyle a \! \left(n +7\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+37 a \! \left(n +2\right)-62 a \! \left(n +3\right)+59 a \! \left(n +4\right)-32 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{\left(\left\{\begin{array}{cc}2122 \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =1\right)^{4}\\+\\2122 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =2\right)^{4}\\+\\2122 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =3\right)^{4}\\+\\2122 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =4\right)^{4}\\+\\2122 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =5\right)^{4}\\-\\6781 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =1\right)^{3}\\-\\6781 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =2\right)^{3}\\-\\6781 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =3\right)^{3}\\-\\6781 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =4\right)^{3}\\-\\6781 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =5\right)^{3}\\+\\12985 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =1\right)^{2}\\+\\12985 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =2\right)^{2}\\+\\12985 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =3\right)^{2}\\+\\12985 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =4\right)^{2}\\+\\12985 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =5\right)^{2}\\-\\9897 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =1\right)\\-\\9897 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =2\right)\\-\\9897 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =3\right)\\-\\9897 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =4\right)\\-\\9897 \\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z \\-1, \mathit{index} =5\right)\\+18140 & n =0 \\ 33952 \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{2}\right. \\-\\2 \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)\\ \left. +3\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}\right. \\-\\2 \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)\\ \left. +3\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}\right. \\-\\2 \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)\\ \left. +3\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}\right. \\-\\\frac{3 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)}{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}\right. \\-\\\frac{3 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{2}\right. \\-\\\frac{3 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)}{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}\right. \\-\\\frac{3 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)}{2}\\ \left. +1\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}\right. \\-\\2 \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)\\ \left. +3\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{2}\right. \\-\\2 \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)\\ \left. +3\right)\\ \\\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{2}\right. \\-\\\frac{3 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)}{2}\\ \left. +1\right)\\ \\\left(-\frac{2738 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)}{1061}\right. \\+\\\frac{3896 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)}{1061}\\-\\\frac{1869 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)}{1061}\\+\\\frac{646 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)}{1061}\\-\\\frac{2738 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{3896 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{1869 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{646 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{2738 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{3896 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{1869 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{646 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{2738 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{3896 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{1869 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{646 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{2738 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{3896 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\-\\\frac{1869 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\frac{646 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4} \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)}{1061}\\+\\\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n} \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)\\+\\ \left. \left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n} \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)+\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n} \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right) \left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n} \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)+\left(\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}-\frac{1797 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right) \left(\left(\sqrt{5}+\frac{5}{3}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}-\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \left(\sqrt{5}-\frac{5}{3}\right)\right)}{5305}\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)\right)\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)\right) \mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)\right) & \text{otherwise} \end{array}\right.\right)}{2396}\)
This specification was found using the strategy pack "All The Strategies 1 Expand Verified" and has 128 rules.
Found on January 21, 2022.Finding the specification took 0 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{40}\! \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_{14}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)+F_{39}\! \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_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \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_{45}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{64}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{12}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{12}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{80}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{77}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{77}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{91}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{12}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{20}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{109}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{101}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{127}\! \left(x \right)+F_{20}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{12}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{122}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{12}\! \left(x \right) F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{125}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{114}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{100}\! \left(x \right) F_{12}\! \left(x \right)\\
\end{align*}\)