Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 31245, 31254, 32145, 32154)
Generating Function
\(\displaystyle -\frac{\left(x +1\right) \left(4 x -1\right)}{4 x^{3}-2 x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 424, 1800, 7648, 32496, 138080, 586720, 2493056, 10593344, 45012608, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{3}-2 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x +1\right) \left(4 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 3 \right)} = - 4 a{\left(n \right)} + 2 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)}, \quad n \geq 3\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 3 \right)} = - 4 a{\left(n \right)} + 2 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)}, \quad n \geq 3\)
Explicit Closed Form
\(\displaystyle -\frac{560 \left(\left(\left(\left(-\frac{201 i \sqrt{79}}{280}+\frac{7663 i}{560}\right) \sqrt{3}+\frac{359 \sqrt{79}}{280}+\frac{4661}{560}\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{189 i \sqrt{79}}{1040}-\frac{4187 i}{3640}\right) \sqrt{3}-\frac{821 \sqrt{79}}{1040}+\frac{9559}{3640}\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{533 \sqrt{79}}{280}-\frac{4108}{35}\right) \left(\frac{\left(-169 i \sqrt{3}-169\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{1}{3}}}{2028}+\frac{1}{6}+\frac{\left(\left(3 i \sqrt{79}-8 i\right) \sqrt{3}+9 \sqrt{79}+8\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}}}{2028}\right)^{-n}+\left(\left(\left(i \sqrt{79}+\frac{1501 i}{560}\right) \sqrt{3}-\frac{61 \sqrt{79}}{140}-\frac{395}{16}\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{101 i \sqrt{79}}{208}+\frac{1343 i}{1820}\right) \sqrt{3}-\frac{127 \sqrt{79}}{1040}-\frac{79}{26}\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{533 \sqrt{79}}{280}-\frac{4108}{35}\right) \left(\frac{\left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}-\frac{4 \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}}}{507}-\frac{i \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{79}\, \sqrt{3}}{338}\right)^{-n}-\frac{3081 \left(\frac{\left(169 i \sqrt{3}-169\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{1}{3}}}{2028}+\frac{1}{6}+\frac{\left(\left(3 i \sqrt{79}+8 i\right) \sqrt{3}-9 \sqrt{79}+8\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}}}{2028}\right)^{-n}}{140}\right) \left(\left(\left(i \sqrt{79}+\frac{395 i}{16}\right) \sqrt{3}+3 \sqrt{79}-\frac{395}{16}\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{1027}{4}+\left(\left(\frac{101 i \sqrt{79}}{208}-\frac{79 i}{26}\right) \sqrt{3}-\frac{303 \sqrt{79}}{208}-\frac{79}{26}\right) \left(-8+3 i \sqrt{79}\, \sqrt{3}\right)^{\frac{2}{3}}\right)}{9492561}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 84 rules.
Finding the specification took 706 seconds.
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Copy 84 equations to clipboard:
\(\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_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= x^{2}\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{15}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \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_{15}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{15}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{15}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{15}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 0\\
F_{65}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{15}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 0\\
F_{71}\! \left(x \right) &= 0\\
F_{72}\! \left(x \right) &= F_{15}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{15}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{15}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{76}\! \left(x \right)\\
\end{align*}\)