0132_1032_2103
Counting sequence:
1, 1, 2, 6, 21, 76, 273, 971, 3439, 12172, 43098, 152649, 540730, 1915445, 6785029, 24034177, 85134498, 301565746, 1068215101, 3783864272, 13403320805, 47477655647, 168176809999, 595721056436, 2110181402286, 7474749309041, 26477286366134, 93788656245849, 332221056144025, 1176803619530785, 4168509892130818, 14765823653424702, 52303953644404325, 185272669581520260, 656278535409667481, 2324689966482025299, 8234588133967308479, 29168810772082027564, 103322656581683975378, 365992684669694072713, 1296430518371668572978, 4592255964029131415885, 16266829992284450715517, 57620864357422032363009, 204106393862308845206178, 722991931482712553676234, 2561004205197662478140685, 9071667681808613501310088, 32133939632800872274295469, 113825827019120229922705671, 403197337290117287924692351, 1428217980533826903013117396, 5059077556487921204107402630, 17920419761830328854168774705, 63478260780635202205193586734, 224854643210811225407391563505, 796487016999133421968696515633, 2821340752360659732524260488385, 9993839787773557579645495909314, 35400485964028742248343140067702, 125396687669793483172692184828789, 444183994946668075350774269055100, 1573402176988353175095096998207073, 5573353472245047273944218958926299, 19742103691531791848469097488712975, 69931085496035228186139039241407500, 247712036927020383696355201327950922, 877453178415375974779935006627659785, 3108141573831055298774753865484206570, 11009754458264440734259694631918359461, 38999090083874801802104639524460908021, 138143773608729381395379981983229472001, 489337113912576495773306413499991235682, 1733344940543587672950115291725077963618, 6139907637262980478284691051142169908893, 21748969239958553134184950911260283501824, 77039866223708039716573759560764948924853, 272892978158358985037946860049189101760431, 966649881139347309654727067448353137407215, 3424096871281450768595818894884634691740980, 12128941008196622055762813893674464363322462, 42963507024045751251465347959574580105459505, 152186652945035525268148278189661582680244582, 539080232012952316782897706833867751217847753, 1909546539880180704238438175628249369390253769, 6764054349299086118320220262324485340406554081, 23959840875699596502148457612259467042292536642, 84871283573931168689501458552850542797540875694, 300633665008692022444980830208900582607286596421, 1064913793342457615036434764442575876290389529908, 3772170316382347717449935517536732472951873277289, 13361897446303634969101787889473198649175879216483, 47330923153216057257149750371660236241168967753823, 167657048374919648187789016292256577006432947476780, 593879941424303175667047093033527123095387524772546, 2103659752123456192963702113595837624857398836908489, 7451648126203277824902257813003207790526702081825890, 26395456651532751733838938393215425196320468500466493, 93498796513615510782587217631124567776278066224312685, 331194306084749677824033933162258344577478295744244865, 1173166633936144777161515599560435580373922958707560418
Generating function in Maple syntax:
-(x^4-7*x^3+8*x^2-5*x+1)/(x^5-5*x^4+13*x^3-12*x^2+6*x-1)
Generating function in latex syntax:
-\frac{x^{4}-7 x^{3}+8 x^{2}-5 x +1}{x^{5}-5 x^{4}+13 x^{3}-12 x^{2}+6 x -1}
Generating function in sympy syntax:
(-x**4 + 7*x**3 - 8*x**2 + 5*x - 1)/(x**5 - 5*x**4 + 13*x**3 - 12*x**2 + 6*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^5-5*x^4+13*x^3-12*x^2+6*x-1)*F(x)+x^4-7*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-5 x^{4}+13 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+x^{4}-7 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
108/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+3)+108/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+3)+108/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+3)+108/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+3)+108/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+3)-941/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+2)-941/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+2)-941/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+2)-941/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+2)-941/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+2)+3514/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+1)+3514/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+1)+3514/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+1)+3514/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+1)+3514/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+1)+573/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n-1)+573/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n-1)+573/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n-1)+573/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n-1)+573/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n-1)-1356/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n)-1356/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n)-1356/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n)-1356/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n)-1356/11497*RootOf(_Z^5-5*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n)
Explicit closed form in latex syntax:
\frac{108 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{11497}+\frac{108 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{11497}+\frac{108 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{11497}+\frac{108 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{11497}+\frac{108 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{11497}-\frac{941 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{11497}-\frac{941 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{11497}-\frac{941 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{11497}-\frac{941 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{11497}-\frac{941 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{11497}+\frac{3514 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{11497}+\frac{3514 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{11497}+\frac{3514 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{11497}+\frac{3514 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{11497}+\frac{3514 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{11497}+\frac{573 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{11497}+\frac{573 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{11497}+\frac{573 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{11497}+\frac{573 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{11497}+\frac{573 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{11497}-\frac{1356 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{11497}-\frac{1356 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{11497}-\frac{1356 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{11497}-\frac{1356 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{11497}-\frac{1356 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{11497}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+5) = a(n)-5*a(n+1)+13*a(n+2)-12*a(n+3)+6*a(n+4), n >= 5
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) = 21
a \! \left(n +5\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+13 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14711/
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[28,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[27,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[22,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[21,x]*F[4,x]
F[27,x] = F[13,x]*F[4,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[51,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[11,x]+F[14,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[34,x]
F[37,x] = 2*F[15,x]+F[38,x]+F[49,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[29,x]+F[44,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[43,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[29,x]+F[37,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[2,x]+F[54,x]
F[54,x] = F[15,x]+F[30,x]+F[55,x]
F[55,x] = F[4,x]*F[53,x]
F[56,x] = F[29,x]+F[57,x]
F[57,x] = F[15,x]+F[38,x]+F[58,x]+F[92,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[73,x]
F[60,x] = F[19,x]+F[61,x]
F[61,x] = 2*F[15,x]+F[26,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[19,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[64,x]
F[67,x] = F[68,x]+F[71,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[22,x]+F[68,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[67,x]
F[73,x] = F[74,x]+F[77,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[34,x]+F[74,x]
F[77,x] = 3*F[15,x]+F[78,x]+F[89,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[83,x]
F[80,x] = F[74,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[80,x]
F[83,x] = F[84,x]+F[87,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[44,x]+F[84,x]
F[87,x] = F[88,x]
F[88,x] = F[4,x]*F[83,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[44,x]+F[91,x]
F[91,x] = F[89,x]
F[92,x] = F[4,x]*F[56,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_{28}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{51}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \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_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{29}\! \left(x \right)+F_{37}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{29}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{58}\! \left(x \right)+F_{92}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{73}\! \left(x \right)
F_{60}\! \left(x \right) = F_{19}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{19}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{22}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{77}\! \left(x \right)
F_{74}\! \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_{34}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{78}\! \left(x \right)+F_{89}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{83}\! \left(x \right)
F_{80}\! \left(x \right) = F_{74}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{87}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{44}\! \left(x \right)+F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{44}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \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_28(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_27(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_22(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_21(x)*F_4(x))
Eq(F_27(x), F_13(x)*F_4(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_51(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_11(x) + F_14(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_34(x))
Eq(F_37(x), 2*F_15(x) + F_38(x) + F_49(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_29(x) + F_44(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_43(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_29(x) + F_37(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_2(x) + F_54(x))
Eq(F_54(x), F_15(x) + F_30(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_53(x))
Eq(F_56(x), F_29(x) + F_57(x))
Eq(F_57(x), F_15(x) + F_38(x) + F_58(x) + F_92(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_73(x))
Eq(F_60(x), F_19(x) + F_61(x))
Eq(F_61(x), 2*F_15(x) + F_26(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_19(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_64(x))
Eq(F_67(x), F_68(x) + F_71(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_22(x) + F_68(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_67(x))
Eq(F_73(x), F_74(x) + F_77(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_34(x) + F_74(x))
Eq(F_77(x), 3*F_15(x) + F_78(x) + F_89(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_83(x))
Eq(F_80(x), F_74(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_80(x))
Eq(F_83(x), F_84(x) + F_87(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_44(x) + F_84(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_4(x)*F_83(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_44(x) + F_91(x))
Eq(F_91(x), F_89(x))
Eq(F_92(x), F_4(x)*F_56(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": [1, 0, 3, 2], "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, 1, 3, 2], "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, 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, 1, 3, 2], "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, 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": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"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, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 0], [0, 1]]}, {"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": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "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, 0], [1, 2]]}, {"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": [0, 2, 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": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"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, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 0], [0, 1]]}, {"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": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"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, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": 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