Av(1243, 1342, 2413, 3124, 3214)
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Generating Function
\(\displaystyle \frac{\left(x -1\right)^{3}}{x^{7}+x^{4}+4 x^{3}-5 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 151, 412, 1133, 3133, 8672, 23986, 66304, 183258, 506540, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 151\)
\(\displaystyle a \! \left(n \right) = -a \! \left(n +3\right)-4 a \! \left(n +4\right)+5 a \! \left(n +5\right)-4 a \! \left(n +6\right)+a \! \left(n +7\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle -\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +5}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +4}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +3}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +2}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n -1}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf} \left(Z^{7}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n}}{56686603}\)
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This specification was found using the strategy pack "Point Placements" and has 59 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{17}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \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_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \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_{33}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{37}\! \left(x \right)\\ \end{align*}\)