0123_0213_0321_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 54, 149, 416, 1173, 3311, 9333, 26295, 74097, 208834, 588592, 1658900, 4675446, 13177321, 37139196, 104673848, 295014812, 831475431, 2343446513, 6604815570, 18615141966, 52465281909, 147869181508, 416757407497, 1174597270004, 3310508037915, 9330400938487, 26296985441451, 74115940772183, 208889824612811, 588739188530148, 1659314103763874, 4676643492731382, 13180743964343688, 37148867927124321, 104701099725533104, 295091637926701972, 831692073937545908, 2344057292543072445, 6606537158290982857, 18619994213762587482, 52478957767678937174, 147907726327124395259, 416866043798093681010, 1174903453573038511373, 3311370992563837320984, 9332833108155945009237, 26303840318795870396643, 74135260697211884869417, 208944276274224482334415, 588892656168960486676817, 1659746639982568147835406, 4677862561327391992964200, 13184179811262213440801688, 37158551585656285356955502, 104728392339158309373679781, 295168559965574289524604840, 831908872524291392483849516, 2344668321942399891632960304, 6608259295563247602066862731, 18624847919308760417392963233, 52492637546816102949732998050, 147946281685594242468626066806, 416974708978417544052558484277, 1175209717653659630063122120416, 3312234173269920155472927061057, 9335265913628405602528977877308, 26310696985569263659097661793595, 74154585661650491050188784321235, 208998742133933826946600889596431, 589046163818783196779827697141279, 1660179288960842667973849672861419, 4679081947713791427537127525259256, 13187616553827205258290606016919442, 37168237768469021392801260772073098, 104755692067231472816858038873377856, 295245502055897913242864247551450357, 832125727624367950129358880351855036, 2345279510619902915942549796619170688, 6609981881748105687860365204695483000, 18629702890078812439936667002396129913, 52506320891884844104957356843991677281, 147984847094355341274872234436233860338, 417083402484679274599306842427895252918, 1175516061568660111052123617449700264419, 3313097578982786223770438194387100088222, 9337699353264406184151998822504856369717, 26317555439681643626195265172801635972160, 74173915663560433820418400411254865371205, 209053222191351420486611992256004839124743, 589199711483705961207901334729047322534685, 1660612050718461883152192563031192531068743, 4680301651959713219323917504968430757101009, 13191054192253724098875668697883684520265242, 37177926476195498116618618454861923189212464, 104782998911571163668174898612181857075206108, 295322464202847026051427995171463863103330470
Generating function in Maple syntax:
(2*x-1)*(x-1)/(2*x^6+3*x^5-x^4-2*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x -1\right)}{2 x^{6}+3 x^{5}-x^{4}-2 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(x - 1)*(2*x - 1)/(2*x**6 + 3*x**5 - x**4 - 2*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^6+3*x^5-x^4-2*x^3+4*x^2-4*x+1)*F(x)-(2*x-1)*(x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{6}+3 x^{5}-x^{4}-2 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right) = 0
Explicit closed form in Maple syntax:
-1003076/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+4)-1003076/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+4)-1003076/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+4)-1003076/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+4)-1003076/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+4)-1003076/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+4)-2383598/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+3)-2383598/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+3)-2383598/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+3)-2383598/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+3)-2383598/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+3)-2383598/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+3)-1102872/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+2)-1102872/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+2)-1102872/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+2)-1102872/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+2)-1102872/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+2)-1102872/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+2)+845941/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+1)+845941/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+1)+845941/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+1)+845941/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+1)+845941/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+1)+845941/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+1)+1025123/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n-1)+1025123/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n-1)+1025123/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n-1)+1025123/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n-1)+1025123/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n-1)+1025123/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n-1)+181758/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n)+181758/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n)+181758/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n)+181758/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n)+181758/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n)+181758/10544351*RootOf(2*_Z^6+3*_Z^5-_Z^4-2*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{1003076 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{10544351}-\frac{1003076 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{10544351}-\frac{1003076 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{10544351}-\frac{1003076 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{10544351}-\frac{1003076 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{10544351}-\frac{1003076 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{10544351}-\frac{2383598 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{10544351}-\frac{2383598 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{10544351}-\frac{2383598 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{10544351}-\frac{2383598 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{10544351}-\frac{2383598 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{10544351}-\frac{2383598 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{10544351}-\frac{1102872 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{10544351}-\frac{1102872 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{10544351}-\frac{1102872 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{10544351}-\frac{1102872 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{10544351}-\frac{1102872 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{10544351}-\frac{1102872 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{10544351}+\frac{845941 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{10544351}+\frac{845941 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{10544351}+\frac{845941 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{10544351}+\frac{845941 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{10544351}+\frac{845941 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{10544351}+\frac{845941 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{10544351}+\frac{1025123 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{10544351}+\frac{1025123 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{10544351}+\frac{1025123 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{10544351}+\frac{1025123 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{10544351}+\frac{1025123 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{10544351}+\frac{1025123 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{10544351}+\frac{181758 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{10544351}+\frac{181758 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{10544351}+\frac{181758 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{10544351}+\frac{181758 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{10544351}+\frac{181758 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{10544351}+\frac{181758 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{10544351}
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) = -2*a(n)-3*a(n+1)+a(n+2)+2*a(n+3)-4*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) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)-4 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/16842/
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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[52,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]
F[29,x] = F[21,x]+F[30,x]
F[30,x] = 2*F[22,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[26,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[22,x]+F[38,x]+F[48,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[11,x]+F[41,x]
F[41,x] = F[27,x]
F[42,x] = F[37,x]+F[43,x]
F[43,x] = 2*F[22,x]+F[35,x]+F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[41,x]
F[47,x] = F[43,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[2,x]*F[4,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[2,x]+F[37,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[49,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,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_{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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \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_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{52}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{22}\! \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_{26}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \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_{37}\! \left(x \right) = F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \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_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{27}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{35}\! \left(x \right)+F_{44}\! \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_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \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_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \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_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_14(x))
Eq(F_10(x), F_1(x) + F_11(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_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_52(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x))
Eq(F_29(x), F_21(x) + F_30(x))
Eq(F_30(x), 2*F_22(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_26(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_22(x) + F_38(x) + F_48(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_11(x) + F_41(x))
Eq(F_41(x), F_27(x))
Eq(F_42(x), F_37(x) + F_43(x))
Eq(F_43(x), 2*F_22(x) + F_35(x) + F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_41(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_2(x)*F_4(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_2(x) + F_37(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_49(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [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, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 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