Av(12453, 12543, 13452, 13542, 21453, 21543)
Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3298, 18940, 111686, 671940, 4107338, 25434224, 159209850, 1005776060, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(2 x -1\right) F \left(x
\right)^{4}+\left(-3 x^{3}+3 x^{2}-4 x +2\right) F \left(x
\right)^{3}+\left(2 x^{3}-4 x^{2}+12 x -7\right) F \left(x
\right)^{2}+\left(-10 x +8\right) F \! \left(x \right)+\left(x +3\right) \left(x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a(6) = 596\)
\(\displaystyle a(7) = 3298\)
\(\displaystyle a(8) = 18940\)
\(\displaystyle a(9) = 111686\)
\(\displaystyle a(10) = 671940\)
\(\displaystyle a(11) = 4107338\)
\(\displaystyle a(12) = 25434224\)
\(\displaystyle a(13) = 159209850\)
\(\displaystyle a(14) = 1005776060\)
\(\displaystyle a(15) = 6404024930\)
\(\displaystyle a(16) = 41056032092\)
\(\displaystyle a(17) = 264794054142\)
\(\displaystyle a(18) = 1716901541372\)
\(\displaystyle a(19) = 11185015160586\)
\(\displaystyle a(20) = 73176070840496\)
\(\displaystyle a(21) = 480575525822306\)
\(\displaystyle a(22) = 3167076242941620\)
\(\displaystyle a(23) = 20937502893392210\)
\(\displaystyle a(24) = 138817201300313228\)
\(\displaystyle a(25) = 922806715358155238\)
\(\displaystyle a(26) = 6149453180742777700\)
\(\displaystyle a(27) = 41071457910920468602\)
\(\displaystyle a(28) = 274884349095984891904\)
\(\displaystyle a(29) = 1843326104400993633850\)
\(\displaystyle a(30) = 12383378300167290777884\)
\(\displaystyle a(31) = 83331488789609688429234\)
\(\displaystyle a(32) = 561649479855567058509260\)
\(\displaystyle a(33) = 3791100802782331105125566\)
\(\displaystyle a(34) = 25625385639187567961330876\)
\(\displaystyle a{\left(n + 35 \right)} = \frac{6174 n \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{\left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{1029 \left(n + 1\right) \left(2087 n^{2} + 8502 n + 8280\right) a{\left(n + 1 \right)}}{8 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(2655 n^{2} + 170637 n + 2740274\right) a{\left(n + 33 \right)}}{8 \left(n + 33\right) \left(n + 37\right)} + \frac{\left(n + 36\right) \left(373 n^{2} + 24669 n + 407606\right) a{\left(n + 34 \right)}}{8 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(34464 n^{3} + 3322418 n^{2} + 106659561 n + 1140282942\right) a{\left(n + 32 \right)}}{8 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{147 \left(199499 n^{3} + 1632401 n^{2} + 4309218 n + 3682080\right) a{\left(n + 2 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(607622 n^{3} + 56835203 n^{2} + 1770374371 n + 18364931360\right) a{\left(n + 31 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{49 \left(1981025 n^{3} + 21620739 n^{2} + 77289820 n + 90736788\right) a{\left(n + 3 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{49 \left(3455731 n^{3} + 43149033 n^{2} + 178156214 n + 244117944\right) a{\left(n + 4 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(11505503 n^{3} + 1043320653 n^{2} + 31506139909 n + 316848513000\right) a{\left(n + 30 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{7 \left(54223267 n^{3} + 536016849 n^{2} + 1263930332 n - 184000056\right) a{\left(n + 5 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(106041700 n^{3} + 9315880761 n^{2} + 272540597411 n + 2655286472622\right) a{\left(n + 29 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(116881694 n^{3} + 9948563684 n^{2} + 281980388583 n + 2661511932826\right) a{\left(n + 28 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{7 \left(349014353 n^{3} + 4949195355 n^{2} + 23524762582 n + 39707276856\right) a{\left(n + 6 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(533333103 n^{3} + 46770159791 n^{2} + 1352382682788 n + 12906571024816\right) a{\left(n + 26 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{7 \left(846366619 n^{3} + 18550771011 n^{2} + 141641308043 n + 375295292406\right) a{\left(n + 7 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(1490789582 n^{3} + 123427605951 n^{2} + 3402454283227 n + 31227450083160\right) a{\left(n + 27 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(4852700663 n^{3} - 879458401197 n^{2} - 42151813502492 n - 449624464588575\right) a{\left(n + 20 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(5536615803 n^{3} + 400852146569 n^{2} + 9636533925854 n + 76943316492492\right) a{\left(n + 25 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(13485201745 n^{3} + 949787322453 n^{2} + 22234025445718 n + 173047026828294\right) a{\left(n + 24 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(35059770625 n^{3} + 964335248307 n^{2} + 9028992002900 n + 28559293562712\right) a{\left(n + 8 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(47171503891 n^{3} + 1547747185381 n^{2} + 16962200974590 n + 61807944168352\right) a{\left(n + 9 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(92091635059 n^{3} + 3569969003471 n^{2} + 45807766261636 n + 194560617092368\right) a{\left(n + 11 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(110086405787 n^{3} + 7406784676296 n^{2} + 165535389861361 n + 1229278853734872\right) a{\left(n + 23 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(201574913981 n^{3} + 12788594106540 n^{2} + 268965585328351 n + 1875632791625076\right) a{\left(n + 22 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(225431286743 n^{3} + 13001432216781 n^{2} + 246019178207668 n + 1523835916460730\right) a{\left(n + 21 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(255009320413 n^{3} + 9482654593227 n^{2} + 116309489221562 n + 470396834722752\right) a{\left(n + 10 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(333341291633 n^{3} + 13183071994611 n^{2} + 173232679043635 n + 756281267847039\right) a{\left(n + 12 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(535600871005 n^{3} + 30662993020380 n^{2} + 583359808566059 n + 3686011194721504\right) a{\left(n + 18 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(608926441381 n^{3} + 38523410218620 n^{2} + 806342042409461 n + 5582129512835808\right) a{\left(n + 19 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(1330085162180 n^{3} + 70862564030925 n^{2} + 1255087315626844 n + 7386794196174741\right) a{\left(n + 17 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(1408509499241 n^{3} + 58163542782849 n^{2} + 798119383908022 n + 3638783383970694\right) a{\left(n + 13 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(1638667486703 n^{3} + 81672790165272 n^{2} + 1353039059032075 n + 7447882488360513\right) a{\left(n + 16 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(2371996953842 n^{3} + 103861569388839 n^{2} + 1510968858341605 n + 7301642297104452\right) a{\left(n + 14 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(3142892553401 n^{3} + 146718649331238 n^{2} + 2275984225539511 n + 11728791574900812\right) a{\left(n + 15 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)}, \quad n \geq 35\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a(6) = 596\)
\(\displaystyle a(7) = 3298\)
\(\displaystyle a(8) = 18940\)
\(\displaystyle a(9) = 111686\)
\(\displaystyle a(10) = 671940\)
\(\displaystyle a(11) = 4107338\)
\(\displaystyle a(12) = 25434224\)
\(\displaystyle a(13) = 159209850\)
\(\displaystyle a(14) = 1005776060\)
\(\displaystyle a(15) = 6404024930\)
\(\displaystyle a(16) = 41056032092\)
\(\displaystyle a(17) = 264794054142\)
\(\displaystyle a(18) = 1716901541372\)
\(\displaystyle a(19) = 11185015160586\)
\(\displaystyle a(20) = 73176070840496\)
\(\displaystyle a(21) = 480575525822306\)
\(\displaystyle a(22) = 3167076242941620\)
\(\displaystyle a(23) = 20937502893392210\)
\(\displaystyle a(24) = 138817201300313228\)
\(\displaystyle a(25) = 922806715358155238\)
\(\displaystyle a(26) = 6149453180742777700\)
\(\displaystyle a(27) = 41071457910920468602\)
\(\displaystyle a(28) = 274884349095984891904\)
\(\displaystyle a(29) = 1843326104400993633850\)
\(\displaystyle a(30) = 12383378300167290777884\)
\(\displaystyle a(31) = 83331488789609688429234\)
\(\displaystyle a(32) = 561649479855567058509260\)
\(\displaystyle a(33) = 3791100802782331105125566\)
\(\displaystyle a(34) = 25625385639187567961330876\)
\(\displaystyle a{\left(n + 35 \right)} = \frac{6174 n \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{\left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{1029 \left(n + 1\right) \left(2087 n^{2} + 8502 n + 8280\right) a{\left(n + 1 \right)}}{8 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(2655 n^{2} + 170637 n + 2740274\right) a{\left(n + 33 \right)}}{8 \left(n + 33\right) \left(n + 37\right)} + \frac{\left(n + 36\right) \left(373 n^{2} + 24669 n + 407606\right) a{\left(n + 34 \right)}}{8 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(34464 n^{3} + 3322418 n^{2} + 106659561 n + 1140282942\right) a{\left(n + 32 \right)}}{8 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{147 \left(199499 n^{3} + 1632401 n^{2} + 4309218 n + 3682080\right) a{\left(n + 2 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(607622 n^{3} + 56835203 n^{2} + 1770374371 n + 18364931360\right) a{\left(n + 31 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{49 \left(1981025 n^{3} + 21620739 n^{2} + 77289820 n + 90736788\right) a{\left(n + 3 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{49 \left(3455731 n^{3} + 43149033 n^{2} + 178156214 n + 244117944\right) a{\left(n + 4 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(11505503 n^{3} + 1043320653 n^{2} + 31506139909 n + 316848513000\right) a{\left(n + 30 \right)}}{16 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{7 \left(54223267 n^{3} + 536016849 n^{2} + 1263930332 n - 184000056\right) a{\left(n + 5 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(106041700 n^{3} + 9315880761 n^{2} + 272540597411 n + 2655286472622\right) a{\left(n + 29 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(116881694 n^{3} + 9948563684 n^{2} + 281980388583 n + 2661511932826\right) a{\left(n + 28 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{7 \left(349014353 n^{3} + 4949195355 n^{2} + 23524762582 n + 39707276856\right) a{\left(n + 6 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(533333103 n^{3} + 46770159791 n^{2} + 1352382682788 n + 12906571024816\right) a{\left(n + 26 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{7 \left(846366619 n^{3} + 18550771011 n^{2} + 141641308043 n + 375295292406\right) a{\left(n + 7 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(1490789582 n^{3} + 123427605951 n^{2} + 3402454283227 n + 31227450083160\right) a{\left(n + 27 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(4852700663 n^{3} - 879458401197 n^{2} - 42151813502492 n - 449624464588575\right) a{\left(n + 20 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(5536615803 n^{3} + 400852146569 n^{2} + 9636533925854 n + 76943316492492\right) a{\left(n + 25 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(13485201745 n^{3} + 949787322453 n^{2} + 22234025445718 n + 173047026828294\right) a{\left(n + 24 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(35059770625 n^{3} + 964335248307 n^{2} + 9028992002900 n + 28559293562712\right) a{\left(n + 8 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(47171503891 n^{3} + 1547747185381 n^{2} + 16962200974590 n + 61807944168352\right) a{\left(n + 9 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{3 \left(92091635059 n^{3} + 3569969003471 n^{2} + 45807766261636 n + 194560617092368\right) a{\left(n + 11 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(110086405787 n^{3} + 7406784676296 n^{2} + 165535389861361 n + 1229278853734872\right) a{\left(n + 23 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(201574913981 n^{3} + 12788594106540 n^{2} + 268965585328351 n + 1875632791625076\right) a{\left(n + 22 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(225431286743 n^{3} + 13001432216781 n^{2} + 246019178207668 n + 1523835916460730\right) a{\left(n + 21 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(255009320413 n^{3} + 9482654593227 n^{2} + 116309489221562 n + 470396834722752\right) a{\left(n + 10 \right)}}{128 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(333341291633 n^{3} + 13183071994611 n^{2} + 173232679043635 n + 756281267847039\right) a{\left(n + 12 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{3 \left(535600871005 n^{3} + 30662993020380 n^{2} + 583359808566059 n + 3686011194721504\right) a{\left(n + 18 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(608926441381 n^{3} + 38523410218620 n^{2} + 806342042409461 n + 5582129512835808\right) a{\left(n + 19 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(1330085162180 n^{3} + 70862564030925 n^{2} + 1255087315626844 n + 7386794196174741\right) a{\left(n + 17 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(1408509499241 n^{3} + 58163542782849 n^{2} + 798119383908022 n + 3638783383970694\right) a{\left(n + 13 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(1638667486703 n^{3} + 81672790165272 n^{2} + 1353039059032075 n + 7447882488360513\right) a{\left(n + 16 \right)}}{32 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} + \frac{\left(2371996953842 n^{3} + 103861569388839 n^{2} + 1510968858341605 n + 7301642297104452\right) a{\left(n + 14 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)} - \frac{\left(3142892553401 n^{3} + 146718649331238 n^{2} + 2275984225539511 n + 11728791574900812\right) a{\left(n + 15 \right)}}{64 \left(n + 33\right) \left(n + 35\right) \left(n + 37\right)}, \quad n \geq 35\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 70 rules.
Finding the specification took 2156 seconds.
Copy 70 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_{4}\! \left(x \right) F_{40}\! \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_{40}\! \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_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{40}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{40}\! \left(x \right) F_{58}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= -F_{64}\! \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_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{40}\! \left(x \right) &= x\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{49}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{48}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{40}\! \left(x \right) F_{53}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{40}\! \left(x \right) F_{55}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{40}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{65}\! \left(x \right) &= \frac{F_{66}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{40}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{61}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 66 rules.
Finding the specification took 1156 seconds.
Copy 66 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_{38}\! \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_{38}\! \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_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{38}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{38}\! \left(x \right) F_{56}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= -F_{59}\! \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_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{38}\! \left(x \right) &= x\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{38}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{38}\! \left(x \right) F_{46}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{38}\! \left(x \right) F_{56}\! \left(x \right)}\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{52}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{2}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{38}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{60}\! \left(x \right) &= \frac{F_{61}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= \frac{F_{64}\! \left(x \right)}{F_{38}\! \left(x \right)}\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{58}\! \left(x \right)\\
\end{align*}\)