Av(1243, 1342, 2413, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{4}}{x^{7}-x^{6}-2 x^{4}+9 x^{3}-9 x^{2}+5 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 62, 179, 505, 1431, 4093, 11759, 33785, 96952, 278014, 797123, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-x^{6}-2 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 179\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)-2 a \! \left(n +3\right)+9 a \! \left(n +4\right)-9 a \! \left(n +5\right)+5 a \! \left(n +6\right)-a \! \left(n +7\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +5}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +4}}{3796787}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +3}}{7593574}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +2}}{3796787}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +1}}{7593574}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n -1}}{3796787}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf} \left(Z^{7}-Z^{6}-2 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n}}{122477}\)
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This specification was found using the strategy pack "Point Placements" and has 76 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{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_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{61}\! \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_{34}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \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_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{30}\! \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_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \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_{51}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{65}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{43}\! \left(x \right)\\ \end{align*}\)