Av(1234, 1432, 2341, 3214, 4123)
View Raw Data
Generating Function
\(\displaystyle \frac{9 x^{11}+8 x^{10}-19 x^{9}-23 x^{8}-26 x^{7}-26 x^{6}+3 x^{5}+5 x^{4}+x^{3}-x +1}{\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 43, 67, 131, 295, 662, 1487, 3342, 7510, 16874, 37915, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) F \! \left(x \right)+9 x^{11}+8 x^{10}-19 x^{9}-23 x^{8}-26 x^{7}-26 x^{6}+3 x^{5}+5 x^{4}+x^{3}-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 43\)
\(\displaystyle a \! \left(6\right) = 67\)
\(\displaystyle a \! \left(7\right) = 131\)
\(\displaystyle a \! \left(8\right) = 295\)
\(\displaystyle a \! \left(9\right) = 662\)
\(\displaystyle a \! \left(10\right) = 1487\)
\(\displaystyle a \! \left(11\right) = 3342\)
\(\displaystyle a \! \left(n +2\right) = a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right), \quad n \geq 12\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}6 & n =3 \\ 19 & n =4 \\ 43 & n =5 \\ 67 & n =6 \\ \frac{\left(\left(-40 \,\mathrm{I} \sqrt{3}+10\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-70 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+350 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+3136\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{15288}\\+\\\frac{\left(\left(25 \,\mathrm{I} \sqrt{3}+55\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-140 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-280 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+3136\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{15288}\\+\\\frac{\left(\left(15 \,\mathrm{I} \sqrt{3}-65\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+210 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-70 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+3136\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}\right)^{-n}}{15288}\\+\frac{5 \cos \left(\frac{n \pi}{2}\right)}{13}+\frac{\sin \left(\frac{n \pi}{2}\right)}{13} & \text{otherwise} \end{array}\right.\)

This specification was found using the strategy pack "Point Placements" and has 110 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_{29}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \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_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \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_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \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_{51}\! \left(x \right)+F_{54}\! \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) &= x^{2}\\ F_{54}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= x^{2}\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{18}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{72}\! \left(x \right)\\ \end{align*}\)