Av(12453, 12534, 12543, 13452, 13542, 14352, 21453, 21534, 21543)
Counting Sequence
1, 1, 2, 6, 24, 111, 549, 2827, 14977, 81094, 446792, 2496848, 14119216, 80642088, 464534988, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{3}-2 x^{2}+x -6\right) x F \left(x
\right)^{3}-\left(x^{4}-5 x^{3}+4 x^{2}-9 x +13\right) x F \left(x
\right)^{2}+\left(-2 x^{4}+2 x^{3}-2 x^{2}+11 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 111\)
\(\displaystyle a \! \left(6\right) = 549\)
\(\displaystyle a \! \left(7\right) = 2827\)
\(\displaystyle a \! \left(8\right) = 14977\)
\(\displaystyle a \! \left(9\right) = 81094\)
\(\displaystyle a \! \left(10\right) = 446792\)
\(\displaystyle a \! \left(11\right) = 2496848\)
\(\displaystyle a \! \left(12\right) = 14119216\)
\(\displaystyle a \! \left(13\right) = 80642088\)
\(\displaystyle a \! \left(14\right) = 464534988\)
\(\displaystyle a \! \left(15\right) = 2695750718\)
\(\displaystyle a \! \left(16\right) = 15744889634\)
\(\displaystyle a \! \left(17\right) = 92483547305\)
\(\displaystyle a \! \left(18\right) = 545980210521\)
\(\displaystyle a \! \left(19\right) = 3237753444070\)
\(\displaystyle a \! \left(20\right) = 19278237886512\)
\(\displaystyle a \! \left(21\right) = 115206870514449\)
\(\displaystyle a \! \left(22\right) = 690766103394039\)
\(\displaystyle a \! \left(23\right) = 4154306963585431\)
\(\displaystyle a \! \left(24\right) = 25053614927602931\)
\(\displaystyle a \! \left(25\right) = 151477870581133357\)
\(\displaystyle a \! \left(26\right) = 918013398802048225\)
\(\displaystyle a \! \left(27\right) = 5575625230010963146\)
\(\displaystyle a \! \left(n +28\right) = -\frac{n \left(2 n +5\right) a \! \left(n \right)}{27 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(32746 n^{2}+1371861 n +14365529\right) a \! \left(n +21\right)}{60 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(17167 n^{2}+736274 n +7864279\right) a \! \left(n +22\right)}{60 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(25759 n^{2}+1158082 n +12984285\right) a \! \left(n +23\right)}{60 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(4277 n^{2}+194540 n +2200569\right) a \! \left(n +24\right)}{20 \left(2 n +57\right) \left(n +28\right)}-\frac{3 \left(121 n^{2}+6155 n +78212\right) a \! \left(n +25\right)}{8 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(877 n^{2}+37159 n +368820\right) a \! \left(n +26\right)}{120 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(803 n^{2}+44663 n +620940\right) a \! \left(n +27\right)}{60 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(102161 n^{2}+4153436 n +42232428\right) a \! \left(n +18\right)}{360 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(24471 n^{2}+953004 n +9274420\right) a \! \left(n +19\right)}{40 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(101027 n^{2}+4302629 n +45691332\right) a \! \left(n +20\right)}{180 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(231008 n^{2}+7542587 n +62210946\right) a \! \left(n +14\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(188500 n^{2}+5564941 n +41319006\right) a \! \left(n +15\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(104554 n^{2}+4046151 n +38701166\right) a \! \left(n +16\right)}{360 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(341639 n^{2}+13505816 n +131406768\right) a \! \left(n +17\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(12930 n^{2}-23879 n -1768964\right) a \! \left(n +12\right)}{360 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(241910 n^{2}+5366075 n +29417622\right) a \! \left(n +13\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(92 n^{2}+446 n +399\right) a \! \left(n +1\right)}{135 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(362 n^{2}+2705 n +4362\right) a \! \left(n +2\right)}{135 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(1796 n^{2}+18962 n +44127\right) a \! \left(n +3\right)}{270 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(5810 n^{2}+89513 n +288090\right) a \! \left(n +4\right)}{540 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(1658 n^{2}+60866 n +294813\right) a \! \left(n +5\right)}{270 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(7354 n^{2}-49523 n -690672\right) a \! \left(n +6\right)}{540 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(5054 n^{2}+45063 n +30839\right) a \! \left(n +7\right)}{90 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(9932 n^{2}+151371 n +526392\right) a \! \left(n +8\right)}{90 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(11424 n^{2}+230249 n +1084139\right) a \! \left(n +9\right)}{90 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(72704 n^{2}+1879019 n +11067144\right) a \! \left(n +10\right)}{540 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(2679 n^{2}-339901 n -3762776\right) a \! \left(n +11\right)}{360 \left(2 n +57\right) \left(n +28\right)}, \quad n \geq 28\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 111\)
\(\displaystyle a \! \left(6\right) = 549\)
\(\displaystyle a \! \left(7\right) = 2827\)
\(\displaystyle a \! \left(8\right) = 14977\)
\(\displaystyle a \! \left(9\right) = 81094\)
\(\displaystyle a \! \left(10\right) = 446792\)
\(\displaystyle a \! \left(11\right) = 2496848\)
\(\displaystyle a \! \left(12\right) = 14119216\)
\(\displaystyle a \! \left(13\right) = 80642088\)
\(\displaystyle a \! \left(14\right) = 464534988\)
\(\displaystyle a \! \left(15\right) = 2695750718\)
\(\displaystyle a \! \left(16\right) = 15744889634\)
\(\displaystyle a \! \left(17\right) = 92483547305\)
\(\displaystyle a \! \left(18\right) = 545980210521\)
\(\displaystyle a \! \left(19\right) = 3237753444070\)
\(\displaystyle a \! \left(20\right) = 19278237886512\)
\(\displaystyle a \! \left(21\right) = 115206870514449\)
\(\displaystyle a \! \left(22\right) = 690766103394039\)
\(\displaystyle a \! \left(23\right) = 4154306963585431\)
\(\displaystyle a \! \left(24\right) = 25053614927602931\)
\(\displaystyle a \! \left(25\right) = 151477870581133357\)
\(\displaystyle a \! \left(26\right) = 918013398802048225\)
\(\displaystyle a \! \left(27\right) = 5575625230010963146\)
\(\displaystyle a \! \left(n +28\right) = -\frac{n \left(2 n +5\right) a \! \left(n \right)}{27 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(32746 n^{2}+1371861 n +14365529\right) a \! \left(n +21\right)}{60 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(17167 n^{2}+736274 n +7864279\right) a \! \left(n +22\right)}{60 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(25759 n^{2}+1158082 n +12984285\right) a \! \left(n +23\right)}{60 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(4277 n^{2}+194540 n +2200569\right) a \! \left(n +24\right)}{20 \left(2 n +57\right) \left(n +28\right)}-\frac{3 \left(121 n^{2}+6155 n +78212\right) a \! \left(n +25\right)}{8 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(877 n^{2}+37159 n +368820\right) a \! \left(n +26\right)}{120 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(803 n^{2}+44663 n +620940\right) a \! \left(n +27\right)}{60 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(102161 n^{2}+4153436 n +42232428\right) a \! \left(n +18\right)}{360 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(24471 n^{2}+953004 n +9274420\right) a \! \left(n +19\right)}{40 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(101027 n^{2}+4302629 n +45691332\right) a \! \left(n +20\right)}{180 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(231008 n^{2}+7542587 n +62210946\right) a \! \left(n +14\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(188500 n^{2}+5564941 n +41319006\right) a \! \left(n +15\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(104554 n^{2}+4046151 n +38701166\right) a \! \left(n +16\right)}{360 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(341639 n^{2}+13505816 n +131406768\right) a \! \left(n +17\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(12930 n^{2}-23879 n -1768964\right) a \! \left(n +12\right)}{360 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(241910 n^{2}+5366075 n +29417622\right) a \! \left(n +13\right)}{1080 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(92 n^{2}+446 n +399\right) a \! \left(n +1\right)}{135 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(362 n^{2}+2705 n +4362\right) a \! \left(n +2\right)}{135 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(1796 n^{2}+18962 n +44127\right) a \! \left(n +3\right)}{270 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(5810 n^{2}+89513 n +288090\right) a \! \left(n +4\right)}{540 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(1658 n^{2}+60866 n +294813\right) a \! \left(n +5\right)}{270 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(7354 n^{2}-49523 n -690672\right) a \! \left(n +6\right)}{540 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(5054 n^{2}+45063 n +30839\right) a \! \left(n +7\right)}{90 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(9932 n^{2}+151371 n +526392\right) a \! \left(n +8\right)}{90 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(11424 n^{2}+230249 n +1084139\right) a \! \left(n +9\right)}{90 \left(2 n +57\right) \left(n +28\right)}+\frac{\left(72704 n^{2}+1879019 n +11067144\right) a \! \left(n +10\right)}{540 \left(2 n +57\right) \left(n +28\right)}-\frac{\left(2679 n^{2}-339901 n -3762776\right) a \! \left(n +11\right)}{360 \left(2 n +57\right) \left(n +28\right)}, \quad n \geq 28\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 106 rules.
Found on January 25, 2022.Finding the specification took 2438 seconds.
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Copy 106 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_{30}\! \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_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{30}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{30}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{17}\! \left(x \right) F_{30}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= x\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{30}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{28}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{0}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \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_{51}\! \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_{30}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{30}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{30}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{30}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 0\\
F_{61}\! \left(x \right) &= F_{30}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{30}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= 2 F_{60}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{30}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{30}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{30}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{30}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{82}\! \left(x \right) &= 2 F_{60}\! \left(x \right)+F_{69}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{30}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{30}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= \frac{F_{90}\! \left(x \right)}{F_{30}\! \left(x \right)}\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{0}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= -F_{40}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= \frac{F_{95}\! \left(x \right)}{F_{30}\! \left(x \right)}\\
F_{95}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{100}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{100}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{100}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{0}\! \left(x \right) F_{41}\! \left(x \right)\\
\end{align*}\)