0231_0312_1203_2103
Counting sequence:
1, 1, 2, 6, 20, 63, 189, 556, 1633, 4813, 14223, 42068, 124423, 367924, 1087835, 3216304, 9509490, 28116694, 83133222, 245802177, 726769496, 2148856808, 6353576075, 18785772018, 55544354233, 164229373001, 485581089866, 1435729762792, 4245058159709, 12551469879915, 37111245796835, 109727751555933, 324434796672687, 959264506178308, 2836281443435266, 8386104539060104, 24795405801196047, 73313198765020235, 216766975170932158, 640920357030187897, 1895025308844094967, 5603068902162372601, 16566734479103509746, 48983279716133292655, 144830092785495867042, 428222771073723358539, 1266137017104659791910, 3743619102891283269458, 11068852579313351026178, 32727554287775620698546, 96766381337608317376309, 286111588872212563106450, 845953317210343875576056, 2501251409354345136779563, 7395512832112644671585552, 21866498443697988172921797, 64653224873340499018927187, 191161813002917221227298260, 565212931329444106078673652, 1671179262864329936559552205, 4941217678899700418286315541, 14609822352883691175033435865, 43197228507923179123367058027, 127722329929454635534946346497, 377639819175351975585230071090, 1116577133423434318624390453762, 3301411640347134444559599052964, 9761366673883209641338302799255, 28861677888789074883359977960521, 85336047541879548413288796058636, 252315233997487507310206230948453, 746026786346808615022771046859537, 2205796127048313220411102510629767, 6521932781967799080628276326393860, 19283562379550131549768758142028243, 57016192971833223993647254008190944, 168581209063776923225975720975594263, 498448292811211440998384936111907788, 1473774580133770252756565833037333534, 4357546297928892060731470454683640302, 12884066528593733578791917820554861414, 38094642939795613438197893789856142401, 112635387087597966648792020097922592708, 333031876540305433195671635572361546028, 984683709620502743064770158826127685995, 2911439043207168051704541869805559152910, 8608324906256350931441252816737928788381, 25452450349104038755625745022655277012421, 75255898891871401275560152031972273539938, 222511005437357917870778939898439900689860, 657903875573689566219534232755706558278085, 1945240904575098615339062206368856055755807, 5751542614842552689498712993135974349399491, 17005730432948954875711976392764006338430181, 50281270073844858225682478182494384151316607, 148667893461399035318180954103469764194897908, 439570092676455922153927764507451988707968126, 1299687927748550874061470403279100411501947248, 3842819922643472528706698954132047206203769095, 11362162133372212045944483326939625036901540295, 33594790009371678394820334991855703078533188758
Generating function in Maple syntax:
(2*x-1)*(x^2+x-1)*(x-1)^2/(7*x^5-2*x^4-9*x^3+12*x^2-6*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}{7 x^{5}-2 x^{4}-9 x^{3}+12 x^{2}-6 x +1}
Generating function in sympy syntax:
(x - 1)**2*(2*x - 1)*(x**2 + x - 1)/(7*x**5 - 2*x**4 - 9*x**3 + 12*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(7*x^5-2*x^4-9*x^3+12*x^2-6*x+1)*F(x)-(2*x-1)*(x^2+x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(7 x^{5}-2 x^{4}-9 x^{3}+12 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,2/7,0)-180257/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 1)^(-n+3)-180257/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 2)^(-n+3)-180257/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 3)^(-n+3)-180257/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 4)^(-n+3)-180257/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 5)^(-n+3)-60897/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 1)^(-n+2)-60897/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 2)^(-n+2)-60897/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 3)^(-n+2)-60897/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 4)^(-n+2)-60897/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 5)^(-n+2)+200403/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 1)^(-n+1)+200403/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 2)^(-n+1)+200403/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 3)^(-n+1)+200403/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 4)^(-n+1)+200403/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 5)^(-n+1)+55445/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 1)^(-n-1)+55445/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 2)^(-n-1)+55445/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 3)^(-n-1)+55445/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 4)^(-n-1)+55445/200677*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 5)^(-n-1)-2217/2749*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 1)^(-n)-2217/2749*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 2)^(-n)-2217/2749*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 3)^(-n)-2217/2749*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 4)^(-n)-2217/2749*RootOf(7*_Z^5-2*_Z^4-9*_Z^3+12*_Z^2-6*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{180257 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{200677}-\frac{180257 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{200677}-\frac{180257 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{200677}-\frac{180257 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{200677}-\frac{180257 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{200677}-\frac{60897 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{200677}-\frac{60897 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{200677}-\frac{60897 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{200677}-\frac{60897 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{200677}-\frac{60897 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{200677}+\frac{200403 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{200677}+\frac{200403 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{200677}+\frac{200403 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{200677}+\frac{200403 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{200677}+\frac{200403 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{200677}+\frac{55445 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{200677}+\frac{55445 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{200677}+\frac{55445 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{200677}+\frac{55445 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{200677}+\frac{55445 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{200677}-\frac{2217 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{2749}-\frac{2217 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{2749}-\frac{2217 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{2749}-\frac{2217 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{2749}-\frac{2217 \mathit{RootOf}\! \left(7 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{2749}+\left(\left\{\begin{array}{cc}\frac{2}{7} & 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) = 63
a(n+5) = -7*a(n)+2*a(n+1)+9*a(n+2)-12*a(n+3)+6*a(n+4), n >= 6
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) = 63
a \! \left(n +5\right) = -7 a \! \left(n \right)+2 a \! \left(n +1\right)+9 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15779/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[83,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[11,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[11,x]+F[30,x]
F[33,x] = F[34,x]+F[45,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[2,x]+F[34,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[2,x]+F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[34,x]
F[45,x] = 2*F[19,x]+F[46,x]+F[59,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[34,x]+F[49,x]
F[49,x] = 2*F[19,x]+F[50,x]+F[52,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[48,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[39,x]
F[55,x] = F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[39,x]+F[56,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[82,x]
F[61,x] = F[62,x]
F[62,x] = F[19,x]+F[63,x]+F[80,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[69,x]
F[65,x] = F[11,x]+F[66,x]
F[66,x] = F[19,x]+F[31,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[65,x]
F[69,x] = F[39,x]+F[70,x]
F[70,x] = 2*F[19,x]+F[71,x]+F[77,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[39,x]+F[74,x]
F[74,x] = 2*F[19,x]+F[57,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[73,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[62,x]+F[79,x]
F[79,x] = F[77,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[2,x]+F[62,x]
F[82,x] = F[79,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[37,x]+F[85,x]
F[85,x] = F[62,x]+F[86,x]
F[86,x] = F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[34,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{83}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \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_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{45}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \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_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{34}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{46}\! \left(x \right)+F_{59}\! \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_{34}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{39}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{39}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{82}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{80}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{11}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{65}\! \left(x \right)
F_{69}\! \left(x \right) = F_{39}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{71}\! \left(x \right)+F_{77}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{39}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{57}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{62}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{2}\! \left(x \right)+F_{62}\! \left(x \right)
F_{82}\! \left(x \right) = F_{79}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{37}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{62}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{34}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_83(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_11(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_11(x) + F_30(x))
Eq(F_33(x), F_34(x) + F_45(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_2(x) + F_34(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_2(x) + F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_34(x))
Eq(F_45(x), 2*F_19(x) + F_46(x) + F_59(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_34(x) + F_49(x))
Eq(F_49(x), 2*F_19(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_48(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_39(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_39(x) + F_56(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_82(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_19(x) + F_63(x) + F_80(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_69(x))
Eq(F_65(x), F_11(x) + F_66(x))
Eq(F_66(x), F_19(x) + F_31(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_65(x))
Eq(F_69(x), F_39(x) + F_70(x))
Eq(F_70(x), 2*F_19(x) + F_71(x) + F_77(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_39(x) + F_74(x))
Eq(F_74(x), 2*F_19(x) + F_57(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_73(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_62(x) + F_79(x))
Eq(F_79(x), F_77(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_2(x) + F_62(x))
Eq(F_82(x), F_79(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_37(x) + F_85(x))
Eq(F_85(x), F_62(x) + F_86(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_34(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 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