0132_0231_0321_1203_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 145, 389, 1058, 2896, 7922, 21630, 59018, 161059, 439634, 1200125, 3276032, 8942450, 24409697, 66630098, 181877980, 496466511, 1355187751, 3699208320, 10097598754, 27563062853, 75237934182, 205374371537, 560603269333, 1530259212154, 4177095260674, 11402071433760, 31123837243900, 84957654420780, 231905949977847, 633025593744020, 1727947913319431, 4716719230212132, 12875064187540620, 35144614241724937, 95933029319642740, 261865048543257436, 714803901585724849, 1951175311729155361, 5326055284051404126, 14538347588884628854, 39684821005920940903, 108326273577066408922, 295694455704935970505, 807146855951199670389, 2203240657721369372642, 6014109278931483859808, 16416504612045835878258, 44811560811080598995522, 122320556645884385761778, 333894162741576806399311, 911419265656600675416250, 2487869422422159491973013, 6791050503594699767296044, 18537294010179016132626122, 50600605758700343145171481, 138122710992309072996896942, 377028753031966580392141632, 1029162254285284486038124807, 2809268357195467961424628151, 7668361980706745450795554436, 20932060590271737104461649426, 57137516676596101842216424361, 155966289027730920544734029770, 425735746458256114389033396457, 1162116037653123785893570601261, 3172187668537755873095010762338, 8659010183479292102066379584858, 23636198482594813278366829875112, 64518907689300952846475267800544, 176115014962573289375144846930668, 480735021811456058824748166441659, 1312245643820739365112374941843388, 3581991225099197171292380381228503, 9777636677329593618156942230718720, 26689674258264927893218494835816364, 72853874153855242184171271643056929, 198866682592881705490575638588964232, 542839456443719979696072362391484568, 1481769955781728497171865515417213969, 4044735834498101632842198638643811809, 11040774519039405499706942299165206602, 30137617626491010855325563517270086646, 82265605065504549266121554642266325979, 224557556627993417942660430940210810366, 612966940419174544231029082763696138977, 1673194506071702787998076817502256876229, 4567260761622886235518892628777221084226, 12467092611745727036830516071433058909408, 34030988441880964698742155509036376724050, 92893203764311303378715999497887446612950, 253567342609955307071765071460611406538202, 692153942729839708201762457014307480933211, 1889348507995339862389481215554548774224050, 5157288811482666748479316190906521329362445, 14077671627277089553771215430364258953263576
Generating function in Maple syntax:
-(x-1)^3/(x^6+x^5-x^4-4*x^3+5*x^2-4*x+1)
Generating function in latex syntax:
-\frac{\left(x -1\right)^{3}}{x^{6}+x^{5}-x^{4}-4 x^{3}+5 x^{2}-4 x +1}
Generating function in sympy syntax:
-(x - 1)**3/(x**6 + x**5 - x**4 - 4*x**3 + 5*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6+x^5-x^4-4*x^3+5*x^2-4*x+1)*F(x)+(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+x^{5}-x^{4}-4 x^{3}+5 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
-203804/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+4)-203804/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+4)-203804/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+4)-203804/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+4)-203804/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+4)-203804/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 6)^(-n+4)-437224/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+3)-437224/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 6)^(-n+3)-437224/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+3)-437224/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+3)-437224/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+3)-437224/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+3)-245407/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+2)-245407/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+2)-245407/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+2)-245407/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+2)-245407/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+2)-245407/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 6)^(-n+2)+607518/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+1)+607518/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+1)+607518/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+1)+607518/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+1)+607518/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+1)+607518/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 6)^(-n+1)+496503/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n-1)+496503/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n-1)+496503/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n-1)+496503/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n-1)+496503/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n-1)+496503/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 6)^(-n-1)+436316/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n)+436316/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n)+436316/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n)+436316/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n)+436316/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n)+436316/5672693*RootOf(_Z^6+_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{203804 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{5672693}-\frac{203804 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{5672693}-\frac{203804 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{5672693}-\frac{203804 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{5672693}-\frac{203804 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{5672693}-\frac{203804 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{5672693}-\frac{437224 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{5672693}-\frac{437224 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{5672693}-\frac{437224 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{5672693}-\frac{437224 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{5672693}-\frac{437224 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{5672693}-\frac{437224 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{5672693}-\frac{245407 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{5672693}-\frac{245407 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{5672693}-\frac{245407 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{5672693}-\frac{245407 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{5672693}-\frac{245407 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{5672693}-\frac{245407 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{5672693}+\frac{607518 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{5672693}+\frac{607518 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{5672693}+\frac{607518 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{5672693}+\frac{607518 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{5672693}+\frac{607518 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{5672693}+\frac{607518 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{5672693}+\frac{496503 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{5672693}+\frac{496503 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{5672693}+\frac{496503 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{5672693}+\frac{496503 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{5672693}+\frac{496503 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{5672693}+\frac{496503 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{5672693}+\frac{436316 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{5672693}+\frac{436316 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{5672693}+\frac{436316 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{5672693}+\frac{436316 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{5672693}+\frac{436316 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{5672693}+\frac{436316 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{5672693}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(n+6) = -a(n)-a(n+1)+a(n+2)+4*a(n+3)-5*a(n+4)+4*a(n+5), 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) = 19
a \! \left(5\right) = 54
a \! \left(n +6\right) = -a \! \left(n \right)-a \! \left(n +1\right)+a \! \left(n +2\right)+4 a \! \left(n +3\right)-5 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18664/
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[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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[14,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[58,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[30,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[16,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[14,x]
F[30,x] = F[31,x]+F[41,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = F[35,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[35,x]+F[38,x]
F[41,x] = 2*F[19,x]+F[42,x]+F[44,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[40,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[56,x]
F[46,x] = F[47,x]
F[47,x] = F[19,x]+F[36,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[19,x]+F[24,x]+F[52,x]
F[52,x] = F[28,x]*F[4,x]
F[53,x] = F[35,x]+F[54,x]
F[54,x] = 2*F[19,x]+F[42,x]+F[55,x]
F[55,x] = F[4,x]*F[46,x]
F[56,x] = F[57,x]
F[57,x] = F[39,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[34,x]+F[60,x]
F[60,x] = F[47,x]+F[57,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_{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_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)+F_{14}\! \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_{58}\! \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_{30}\! \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_{16}\! \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_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{39}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{34}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{47}\! \left(x \right)+F_{57}\! \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_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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_58(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_30(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_16(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_14(x))
Eq(F_30(x), F_31(x) + F_41(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), F_35(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_35(x) + F_38(x))
Eq(F_41(x), 2*F_19(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_56(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_19(x) + F_36(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_19(x) + F_24(x) + F_52(x))
Eq(F_52(x), F_28(x)*F_4(x))
Eq(F_53(x), F_35(x) + F_54(x))
Eq(F_54(x), 2*F_19(x) + F_42(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_46(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_39(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_34(x) + F_60(x))
Eq(F_60(x), F_47(x) + F_57(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, 2, 3, 1], "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, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 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, 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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [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": [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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [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": []}], "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, 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": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [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": [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, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"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]]}], "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": [1, 2, 0], "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, 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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [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": [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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [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, 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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [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": [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, 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