Av(1243, 1432, 2143)
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Counting Sequence
1, 1, 2, 6, 21, 79, 313, 1290, 5475, 23764, 105001, 470738, 2136022, 9791501, 45275765, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x +1\right) x F \left(x \right)^{4}+\left(x^{2}-7 x -1\right) F \left(x \right)^{3}+\left(-x^{2}+7 x +6\right) F \left(x \right)^{2}-10 F \! \left(x \right)+5 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 79\)
\(\displaystyle a \! \left(6\right) = 313\)
\(\displaystyle a \! \left(7\right) = 1290\)
\(\displaystyle a \! \left(8\right) = 5475\)
\(\displaystyle a \! \left(9\right) = 23764\)
\(\displaystyle a \! \left(10\right) = 105001\)
\(\displaystyle a \! \left(11\right) = 470738\)
\(\displaystyle a \! \left(12\right) = 2136022\)
\(\displaystyle a \! \left(n +13\right) = -\frac{16 n \left(n +1\right) \left(n +2\right) a \! \left(n \right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}+\frac{2 \left(237 n +628\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}-\frac{\left(n +2\right) \left(4075 n^{2}+26467 n +41536\right) a \! \left(n +2\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}+\frac{2 \left(4211 n^{3}+52076 n^{2}+204311 n +257406\right) a \! \left(n +3\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}-\frac{2 \left(3064 n^{3}+50253 n^{2}+259334 n +426528\right) a \! \left(n +4\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}+\frac{\left(4417 n^{3}+90167 n^{2}+575174 n +1166476\right) a \! \left(n +5\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}-\frac{\left(841 n^{3}+31655 n^{2}+291840 n +787068\right) a \! \left(n +6\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}-\frac{2 \left(98 n^{3}-2161 n^{2}-45572 n -177626\right) a \! \left(n +7\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}+\frac{2 \left(158 n^{3}+1589 n^{2}-6352 n -71226\right) a \! \left(n +8\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}-\frac{3 \left(75 n^{3}+1597 n^{2}+10040 n +15672\right) a \! \left(n +9\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}+\frac{\left(49 n^{3}+1147 n^{2}+7670 n +10844\right) a \! \left(n +10\right)}{\left(n +14\right) \left(n +13\right) \left(n +12\right)}-\frac{2 \left(14 n^{2}+275 n +1310\right) a \! \left(n +11\right)}{\left(n +13\right) \left(n +14\right)}+\frac{10 \left(n +11\right) a \! \left(n +12\right)}{n +14}, \quad n \geq 13\)
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This specification was found using the strategy pack "Point Placements Req Corrob" and has 47 rules.

Found on July 23, 2021.

Finding the specification took 65 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_{12}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= \frac{F_{11}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{11}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{16}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{2}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{10}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{35}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements" and has 88 rules.

Found on July 23, 2021.

Finding the specification took 33 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_{13}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{28}\! \left(x \right) &= 0\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{13}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \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_{15}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{2}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{27}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= \frac{F_{58}\! \left(x \right)}{F_{67}\! \left(x \right)}\\ F_{58}\! \left(x \right) &= -F_{68}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= -F_{66}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= \frac{F_{61}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= \frac{F_{65}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{11}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{16}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= \frac{F_{74}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{74}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{35}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{11} \left(x \right)^{2}\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{2}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{11}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{2}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\ \end{align*}\)