Av(1324, 2431, 3241, 4132)
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Generating Function
\(\displaystyle \frac{x^{7}-5 x^{6}+20 x^{5}-39 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 193, 555, 1544, 4203, 11284, 30033, 79501, 209724, 552016, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{7}+5 x^{6}-20 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 193\)
\(\displaystyle a \! \left(7\right) = 555\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{3}}{6}+\frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{n}{3}+2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(24 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-24 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}-\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{7 n}{3}\\-2^{n -1}+3 & \text{otherwise} \end{array}\right.\)

This specification was found using the strategy pack "Point Placements" and has 68 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{23}\! \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_{16}\! \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_{20}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{25}\! \left(x \right) &= 0\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{54}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{55}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{16}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{50}\! \left(x \right)\\ \end{align*}\)