Av(1342, 1432, 2143, 2314)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(-1+2 x \right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}{7 x^{6}-8 x^{5}-7 x^{4}+21 x^{3}-18 x^{2}+7 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 197, 596, 1796, 5415, 16342, 49340, 148978, 449807, 1358043, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(7 x^{6}-8 x^{5}-7 x^{4}+21 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)-\left(-1+2 x \right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 197\)
\(\displaystyle a \! \left(n +6\right) = 7 a \! \left(n \right)-8 a \! \left(n +1\right)-7 a \! \left(n +2\right)+21 a \! \left(n +3\right)-18 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{2458876 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +4}}{1637701}+\frac{2458876 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +4}}{1637701}+\frac{2458876 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +4}}{1637701}+\frac{2458876 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +4}}{1637701}+\frac{2458876 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +4}}{1637701}+\frac{2458876 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +4}}{1637701}-\frac{2075830 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{1637701}-\frac{2075830 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{1637701}-\frac{2075830 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{1637701}-\frac{2075830 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{1637701}-\frac{2075830 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{1637701}-\frac{2075830 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +3}}{1637701}-\frac{3620813 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{1637701}-\frac{3620813 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{1637701}-\frac{3620813 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{1637701}-\frac{3620813 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{1637701}-\frac{3620813 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{1637701}-\frac{3620813 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +2}}{1637701}+\frac{480952 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{125977}+\frac{480952 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{125977}+\frac{480952 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{125977}+\frac{480952 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{125977}+\frac{480952 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{125977}+\frac{480952 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +1}}{125977}+\frac{790405 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{1637701}+\frac{790405 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{1637701}+\frac{790405 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{1637701}+\frac{790405 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{1637701}+\frac{790405 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{1637701}+\frac{790405 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n -1}}{1637701}-\frac{3588820 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{1637701}-\frac{3588820 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{1637701}-\frac{3588820 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{1637701}-\frac{3588820 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{1637701}-\frac{3588820 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{1637701}-\frac{3588820 \mathit{RootOf} \left(7 Z^{6}-8 Z^{5}-7 Z^{4}+21 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n}}{1637701}+\left(\left\{\begin{array}{cc}\frac{2}{7} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)

This specification was found using the strategy pack "Point Placements" and has 54 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \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) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= x^{2}\\ 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_{10}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{10}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= 0\\ F_{51}\! \left(x \right) &= F_{10}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{10}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{48}\! \left(x \right)\\ \end{align*}\)