0132_0321_1032_2130
Counting sequence:
1, 1, 2, 6, 20, 65, 205, 636, 1959, 6017, 18461, 56617, 173606, 532296, 1632039, 5003846, 15341790, 47037876, 144217918, 442171435, 1355695399, 4156555206, 12743977065, 39072968619, 119797522239, 367298591307, 1126135604848, 3452726012242, 10586040317121, 32456745539016, 99512216034136, 305103946053942, 935447140135520, 2868076153407013, 8793506836260257, 26960847042885968, 82661819317925867, 253440715793549397, 777047940052021345, 2382425015051420717, 7304502926759926906, 22395568662165319452, 68664699122003084379, 210525616769909513226, 645470465655160273490, 1979009150646216213432, 6067631947135159303106, 18603328556552369696095, 57037710328893767002715, 174877328520703560139738, 536172995973963933943469, 1643903666893342818226303, 5040200245662140006556547, 15453228207940297904297167, 47379518750709841328730692, 145265362475875117818719398, 445382859337954663780718581, 1365541571721860669846358284, 4186743483735372181059617228, 12836534135315907957354811194, 39356748090073677699660189028, 120667588610545008767187059673, 369966210311876233578018230101, 1134314511034905025735506373988, 3477802496773182966024375068847, 10662924690548718241749577843753, 32692472635178710193545396825765, 100234954106846275744429974112817, 307319857293096358933984990726606, 942241112675865123585803782823408, 2888906438512121256281058013442447, 8857372383991667253767894791485646, 27156658486006229365823666348199366, 83262176202330927345770971937562044, 255281406934522230066078676415729990, 782691489688024002617455481432577939, 2399728109408246205247397466703585423, 7357554125674032761068352273092289518, 22558223367051317721552612649029349873, 69163397616346001950713116364468375379, 212054623806221204898128351608345933767, 650158393418349167413668411123003535955, 1993382313222295576095201832001470137624, 6111699990175540971552134099552807186170, 18738440951415342011098478016396607714089, 57451964241391752980914393287324864972624, 176147428900420224464458677338318173178432, 540067117250523410609563318308662585737534, 1655843022836166844176350969815248382640488, 5076806249996805001294630257400518709606029, 15565462030247513678400369401964608240712201, 47723627076615249469660135102708305033721208, 146320396845401439241807430159997793444869747, 448617589325824186197969395277273747220352509, 1375459237341721135986207742021415307297428713, 4217150995866591766100615288803454502770969397, 12929763412189156167515585414466870699128145442, 39642588576753446288280520165502077130114268100, 121543974082636682192009809049356770962348105411, 372653203692746750294212751246428423659181461650, 1142552819015536105617917754166868749109666050874
Generating function in Maple syntax:
(x^6-4*x^5+12*x^4-17*x^3+14*x^2-6*x+1)/(2*x^4-7*x^3+8*x^2-5*x+1)/(-1+x)^2
Generating function in latex syntax:
\frac{x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1}{\left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) \left(-1+x \right)^{2}}
Generating function in sympy syntax:
(x**6 - 4*x**5 + 12*x**4 - 17*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**2*(2*x**4 - 7*x**3 + 8*x**2 - 5*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x^4-7*x^3+8*x^2-5*x+1)*(-1+x)^2*F(x)-x^6+4*x^5-12*x^4+17*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) \left(-1+x \right)^{2} F \! \left(x \right)-x^{6}+4 x^{5}-12 x^{4}+17 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
2-n-634/751*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+1018/751*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))-790/751*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+233/751*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+piecewise(n = 0,1/2,0)
Explicit closed form in latex syntax:
2-n -\frac{634 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{751}+\frac{1018 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{751}-\frac{790 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{751}+\frac{233 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{751}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 205
a(n+4) = -2*a(n)+7*a(n+1)-8*a(n+2)+5*a(n+3)+n, n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 205
a \! \left(n +4\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15425/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[41,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[28,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[21,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[36,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[38,x]
F[41,x] = F[15,x]+F[42,x]+F[67,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[15,x]+F[19,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = F[14,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[15,x]+F[42,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[21,x]+F[52,x]
F[55,x] = 2*F[15,x]+F[56,x]+F[63,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[52,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[58,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = 2*F[15,x]+F[60,x]+F[63,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[36,x]+F[62,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[61,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[54,x]+F[69,x]
F[69,x] = F[70,x]+F[90,x]
F[70,x] = F[15,x]+F[37,x]+F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]+F[80,x]
F[73,x] = F[10,x]+F[74,x]
F[74,x] = F[15,x]+F[30,x]+F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[17,x]+F[77,x]
F[77,x] = F[29,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[77,x]
F[80,x] = F[36,x]+F[81,x]
F[81,x] = 2*F[15,x]+F[40,x]+F[82,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[84,x]+F[87,x]
F[84,x] = F[36,x]+F[85,x]
F[85,x] = F[86,x]
F[86,x] = F[12,x]*F[84,x]
F[87,x] = F[39,x]+F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[12,x]*F[87,x]
F[90,x] = F[113,x]+F[15,x]+F[60,x]+F[91,x]
F[91,x] = F[12,x]*F[92,x]
F[92,x] = F[103,x]+F[93,x]
F[93,x] = F[18,x]+F[94,x]
F[94,x] = 2*F[15,x]+F[50,x]+F[95,x]
F[95,x] = F[12,x]*F[96,x]
F[96,x] = F[100,x]+F[97,x]
F[97,x] = F[18,x]+F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[12,x]*F[97,x]
F[100,x] = F[101,x]+F[49,x]
F[101,x] = F[102,x]
F[102,x] = F[100,x]*F[12,x]
F[103,x] = F[104,x]+F[59,x]
F[104,x] = 3*F[15,x]+F[105,x]+F[66,x]
F[105,x] = F[106,x]*F[12,x]
F[106,x] = F[107,x]+F[110,x]
F[107,x] = F[108,x]+F[59,x]
F[108,x] = F[109,x]
F[109,x] = F[107,x]*F[12,x]
F[110,x] = F[111,x]+F[65,x]
F[111,x] = F[112,x]
F[112,x] = F[110,x]*F[12,x]
F[113,x] = F[12,x]*F[69,x]
System of equations in latex syntax:
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_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \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_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{67}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{36}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{54}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{90}\! \left(x \right)
F_{70}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{80}\! \left(x \right)
F_{73}\! \left(x \right) = F_{10}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{17}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{29}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{36}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{87}\! \left(x \right)
F_{84}\! \left(x \right) = F_{36}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = F_{39}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{113}\! \left(x \right)+F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{103}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{18}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{12}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{100}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{18}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{12}\! \left(x \right) F_{97}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{49}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{100}\! \left(x \right) F_{12}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{59}\! \left(x \right)
F_{104}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{105}\! \left(x \right)+F_{66}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{110}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{59}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = F_{107}\! \left(x \right) F_{12}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{65}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_12(x)*F_4(x))
Eq(F_4(x), F_20(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_41(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_31(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_28(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_21(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_36(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_38(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_67(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_15(x) + F_19(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), F_14(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_15(x) + F_42(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_21(x) + F_52(x))
Eq(F_55(x), 2*F_15(x) + F_56(x) + F_63(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_52(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_58(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), 2*F_15(x) + F_60(x) + F_63(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_36(x) + F_62(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_61(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_54(x) + F_69(x))
Eq(F_69(x), F_70(x) + F_90(x))
Eq(F_70(x), F_15(x) + F_37(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_80(x))
Eq(F_73(x), F_10(x) + F_74(x))
Eq(F_74(x), F_15(x) + F_30(x) + F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_17(x) + F_77(x))
Eq(F_77(x), F_29(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_77(x))
Eq(F_80(x), F_36(x) + F_81(x))
Eq(F_81(x), 2*F_15(x) + F_40(x) + F_82(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_87(x))
Eq(F_84(x), F_36(x) + F_85(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_12(x)*F_84(x))
Eq(F_87(x), F_39(x) + F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_12(x)*F_87(x))
Eq(F_90(x), F_113(x) + F_15(x) + F_60(x) + F_91(x))
Eq(F_91(x), F_12(x)*F_92(x))
Eq(F_92(x), F_103(x) + F_93(x))
Eq(F_93(x), F_18(x) + F_94(x))
Eq(F_94(x), 2*F_15(x) + F_50(x) + F_95(x))
Eq(F_95(x), F_12(x)*F_96(x))
Eq(F_96(x), F_100(x) + F_97(x))
Eq(F_97(x), F_18(x) + F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_12(x)*F_97(x))
Eq(F_100(x), F_101(x) + F_49(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_100(x)*F_12(x))
Eq(F_103(x), F_104(x) + F_59(x))
Eq(F_104(x), F_105(x) + 3*F_15(x) + F_66(x))
Eq(F_105(x), F_106(x)*F_12(x))
Eq(F_106(x), F_107(x) + F_110(x))
Eq(F_107(x), F_108(x) + F_59(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_107(x)*F_12(x))
Eq(F_110(x), F_111(x) + F_65(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_110(x)*F_12(x))
Eq(F_113(x), F_12(x)*F_69(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], 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