Av(1234, 1324, 1342, 1423, 2143)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{3}+x^{2}+2 x -1}{x^{3}+3 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 183, 568, 1763, 5472, 16984, 52715, 163617, 507835, 1576220, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+3 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(-\left(5+\sqrt{5}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}+2 \sqrt{5}\, \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{2}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right)^{n}+\left(\left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{5}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}+10 \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{16}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{16}\right)^{n} \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n}\right) \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}}{120} & n <0 \\ 1 & n =0 \\ -\frac{\left(3 \left(\frac{\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{4}+\mathrm{I} \sqrt{3}-1\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n} \sqrt{5}\, \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n}+\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(5+\sqrt{5}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right)\right) \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}}{120}\\+\\\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(10 \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{128}-\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n}+\left(\left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{5}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right) \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}+2 \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{32}-\frac{\mathrm{I} \sqrt{3}}{8}-\frac{1}{8}\right)^{-n} \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{2}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{2}\right)^{n} \sqrt{5}\, \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n}\right)}{120} & 0<n \end{array}\right.\)

This specification was found using the strategy pack "Point Placements" and has 71 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_{14}\! \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_{11}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \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_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)+F_{59}\! \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_{40}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{35}\! \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_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \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_{2}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{51}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \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_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{49}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{56}\! \left(x \right)\\ \end{align*}\)