0123_0132_1203_2103
Counting sequence:
1, 1, 2, 6, 20, 65, 213, 701, 2307, 7594, 25001, 82310, 270989, 892181, 2937344, 9670676, 31838965, 104824084, 345114510, 1136227682, 3740826049, 12315999481, 40548221511, 133497753913, 439517434710, 1447032401331, 4764094903072, 15684928841015, 51639817793981, 170014847298704, 559743421584520, 1842854921115725, 6067269626263354, 19975398115165218, 65765419115729784, 216520858634803005, 712856130992697985, 2346951082209488657, 7726915913052183819, 25439486140107704298, 83754949886221355621, 275748165344575290034, 907851426025486703265, 2988938151978619786261, 9840543309453989692804, 32398225624419799282296, 106665352776040175280469, 351176561788662906761160, 1156185906108103999492046, 3806534931244869083499010, 12532334208744960042651433, 41260464847045364340818641, 135842687486448244161327439, 447237708347395578711210109, 1472450018980847622341332042, 4847777854886515807705275875, 15960439965625610202960098744, 52546888805882065450129800303, 173001216077034540092447031745, 569575505691642657683921292464, 1875225296332213868747708529416, 6173843286561534871455479038185, 20326272795899967779417506141850, 66920611132558772959768918251966, 220324121363680860068476324731012, 725377692061460645935069623121001, 2388176078423466383883542927314645, 7862641826419502868344577483718165, 25886339390591375540947960684751323, 85226134146572793444170072028209786, 280591775915972257438213509900181897, 923798145956914039281022674484602014, 3041439870030250569982035025433192781, 10013395808916317372382590545746512085, 32967311507304462991666900702715614464, 108538966077010697491201662564044168636, 357345097869332320791708115005425668189, 1176494705879545755561600043920446263836, 3873398015575204010677961991105735567990, 12752468933419932772557872752599768389074, 41985218984445225263563985721978406042745, 138228810622876499968168373885304487788777, 455093591230140622076678185954328112898319, 1498314105760453839367175775186336009959441, 4932930726298627894799888220736400140091046, 16240790537115538262639327657497394499290003, 53469892829485076430655111677059347927605240, 176040041441504876187181498838058869678733031, 579580293709468311269628241450197150910003997, 1908164268229690562009836399676558350548338720, 6282288949551059939774275060751021839312266416, 20683310710071738372916795775854028320004442389, 68096094491155065857920595954285896609858849010, 224194189699538482701696911518694611765749498650, 738119198621020223081316000063167380857168306664, 2430125205756207192243256733813741963957927068933, 8000751811746020548253413458547771116187801313889, 26341041770824291741890964387279713036541528680905, 86723160260221810265664300935019474927057451659235, 285520466159025542581543181537799653668547817653722, 940024975462751233499856740092364184311906006915109
Generating function in Maple syntax:
(x-1)*(x^2+x+1)*(x^3+2*x^2+2*x-1)/(x^6+2*x^5+3*x^4-x^3-x^2-3*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{2}+x +1\right) \left(x^{3}+2 x^{2}+2 x -1\right)}{x^{6}+2 x^{5}+3 x^{4}-x^{3}-x^{2}-3 x +1}
Generating function in sympy syntax:
(x - 1)*(x**2 + x + 1)*(x**3 + 2*x**2 + 2*x - 1)/(x**6 + 2*x**5 + 3*x**4 - x**3 - x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6+2*x^5+3*x^4-x^3-x^2-3*x+1)*F(x)-(x-1)*(x^2+x+1)*(x^3+2*x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+2 x^{5}+3 x^{4}-x^{3}-x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}+x +1\right) \left(x^{3}+2 x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,0)-55/3471*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 1)^(-n+4)-55/3471*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 2)^(-n+4)-55/3471*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 3)^(-n+4)-55/3471*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 4)^(-n+4)-55/3471*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 5)^(-n+4)-55/3471*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 6)^(-n+4)+62/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 1)^(-n+3)+62/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 2)^(-n+3)+62/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 3)^(-n+3)+62/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 4)^(-n+3)+62/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 5)^(-n+3)+62/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 6)^(-n+3)+12/1157*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 1)^(-n+2)+12/1157*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 2)^(-n+2)+12/1157*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 3)^(-n+2)+12/1157*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 4)^(-n+2)+12/1157*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 5)^(-n+2)+12/1157*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 6)^(-n+2)+1402/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 1)^(-n+1)+1402/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 2)^(-n+1)+1402/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 3)^(-n+1)+1402/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 4)^(-n+1)+1402/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 5)^(-n+1)+1402/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 6)^(-n+1)+512/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 1)^(-n-1)+512/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 2)^(-n-1)+512/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 3)^(-n-1)+512/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 4)^(-n-1)+512/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 5)^(-n-1)+512/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 6)^(-n-1)-382/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 1)^(-n)-382/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 2)^(-n)-382/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 3)^(-n)-382/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 4)^(-n)-382/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 5)^(-n)-382/10413*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^3-_Z^2-3*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{55 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{3471}+\frac{62 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{10413}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{1157}+\frac{1402 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{10413}+\frac{512 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{10413}-\frac{382 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{10413}-\frac{382 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{10413}-\frac{382 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{10413}-\frac{382 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{10413}-\frac{382 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{10413}-\frac{382 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{10413}+\left(\left\{\begin{array}{cc}1 & 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) = 213
a(n+6) = -a(n)-2*a(n+1)-3*a(n+2)+a(n+3)+a(n+4)+3*a(n+5), 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) = 213
a \! \left(n +6\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14837/
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[20,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[19,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[4,x]*F[7,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[13,x]+F[22,x]+F[66,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[36,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[13,x]+F[14,x]+F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[18,x]+F[7,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = 2*F[13,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[18,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[37,x]+F[50,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = 2*F[13,x]+F[45,x]+F[49,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[41,x]
F[48,x] = F[49,x]
F[49,x] = F[37,x]*F[4,x]
F[50,x] = 2*F[13,x]+F[51,x]+F[56,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[47,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[21,x]*F[4,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[21,x]+F[54,x]
F[59,x] = F[50,x]+F[60,x]
F[60,x] = 3*F[13,x]+F[61,x]+F[65,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[54,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[50,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[40,x]+F[68,x]
F[68,x] = F[21,x]+F[69,x]
F[69,x] = 2*F[13,x]+F[51,x]+F[55,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_{20}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)+F_{66}\! \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_{36}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{26}\! \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_{18}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \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_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 3 F_{13}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{54}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{40}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{21}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \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_20(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_19(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_4(x)*F_7(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_13(x) + F_22(x) + F_66(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_36(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_13(x) + F_14(x) + F_26(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_18(x) + F_7(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), 2*F_13(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_18(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_50(x))
Eq(F_37(x), F_38(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_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), 2*F_13(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_41(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_37(x)*F_4(x))
Eq(F_50(x), 2*F_13(x) + F_51(x) + F_56(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_47(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_21(x)*F_4(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_21(x) + F_54(x))
Eq(F_59(x), F_50(x) + F_60(x))
Eq(F_60(x), 3*F_13(x) + F_61(x) + F_65(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_54(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_50(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_40(x) + F_68(x))
Eq(F_68(x), F_21(x) + F_69(x))
Eq(F_69(x), 2*F_13(x) + F_51(x) + F_55(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 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": [0, 1, 2], "pos": [[0, 1], [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": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [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": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [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": [0, 1, 2], "pos": [[1, 2], [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": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [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": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [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": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"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, 1, 2], "pos": [[0, 1], [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": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], 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