Av(1234, 1243, 1432, 3214)
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Generating Function
\(\displaystyle -\frac{x^{4}+3 x^{3}+x^{2}-1}{4 x^{10}+16 x^{9}+21 x^{8}+8 x^{7}-8 x^{6}-12 x^{5}-6 x^{4}-5 x^{3}-2 x^{2}-x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 60, 162, 442, 1245, 3534, 9976, 28023, 78711, 221417, 623200, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{10}+16 x^{9}+21 x^{8}+8 x^{7}-8 x^{6}-12 x^{5}-6 x^{4}-5 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+x^{4}+3 x^{3}+x^{2}-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(7\right) = 442\)
\(\displaystyle a \! \left(8\right) = 1245\)
\(\displaystyle a \! \left(9\right) = 3534\)
\(\displaystyle a \! \left(n +10\right) = -4 a \! \left(n \right)-16 a \! \left(n +1\right)-21 a \! \left(n +2\right)-8 a \! \left(n +3\right)+8 a \! \left(n +4\right)+12 a \! \left(n +5\right)+6 a \! \left(n +6\right)+5 a \! \left(n +7\right)+2 a \! \left(n +8\right)+a \! \left(n +9\right), \quad n \geq 10\)
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Explicit Closed Form
\(\displaystyle -\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +8}}{331520349540242}-\frac{31714115169909 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +8}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +7}}{331520349540242}-\frac{95219674867549 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +7}}{331520349540242}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +6}}{1326081398160968}-\frac{250051898450109 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +6}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +5}}{1326081398160968}+\frac{29948286076775 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +5}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +4}}{1326081398160968}-\frac{37020065045133 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +4}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +3}}{1326081398160968}-\frac{55089305377221 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +3}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +2}}{1326081398160968}+\frac{38760982889421 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +2}}{1326081398160968}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +1}}{663040699080484}+\frac{108562421801273 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +1}}{663040699080484}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n -1}}{1326081398160968}+\frac{65559996125697 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n -1}}{1326081398160968}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n}}{331520349540242}+\frac{40468192057291 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+21 Z^{8}+8 Z^{7}-8 Z^{6}-12 Z^{5}-6 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n}}{331520349540242}\)
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This specification was found using the strategy pack "Point Placements" and has 96 rules.

Found on January 18, 2022.

Finding the specification took 3 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_{19}\! \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_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{81}\! \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_{34}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \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_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{48}\! \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_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{75}\! \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_{39}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{49}\! \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_{54}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{65}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= 0\\ F_{90}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= 0\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{53}\! \left(x \right)\\ \end{align*}\)