0132_1203_1302_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 59, 181, 553, 1687, 5141, 15655, 47649, 144987, 441089, 1341759, 4081241, 12413403, 37755265, 114830359, 349245849, 1062191603, 3230521601, 9825197599, 29881969897, 90881755659, 276403738913, 840641728839, 2556688782713, 7775793516899, 23648931887745, 71924742327471, 218748499291209, 665291292608827, 2023385297784929, 6153827759895767, 18715958803160665, 56921825943732115, 173119330695940417, 526516887340358207, 1601323383654126377, 4870188666791963627, 14811959835188224929, 45048389112130278503, 137008024859657713465, 416689680681193497027, 1267300146504161208321, 3854306299714750981327, 11722303585846531759881, 35651655751158101859483, 108429247586056646202081, 329771548734766606328055, 1002951480118569006509849, 3050328857445076969663475, 9277124888883518097375425, 28215005734156094182498911, 85811774457407234960376809, 260983843310458248993651531, 793743829442572978500169249, 2414054673984633641507600007, 7341990895323550345413059065, 22329581383521933916606788131, 67912125181318649529746061185, 206544702626913889569515519727, 628175220100091389975925618121, 1910502192160199200449919947771, 5810510363123410590229609101665, 17671809442830225591687419392279, 53746199467378891969303418196953, 163461130934685093453966501257299, 497142897381304103522716009918529, 1511986727385564392511766493730431, 4598484411287258983356748016908457, 13985611446084851458846387668087467, 42535172466988054155126564797339809, 129364447437362457145558750449568935, 393442868340605130613977349873076409, 1196598398667723004255219957242929795, 3639277371408858252726271979869277953, 11068324845490887549433914799572354319, 33662675960828163021658770197057474697, 102380059192549040847293324743016719195, 311373835296603587062949659669067083745, 946997550811852409692327491377324328247, 2880153242128977223720919127375296955545, 8759560878520317491933741257917419824307, 26640911206441829175357481005924158060481, 81024398340548804186133159582663491069599, 246423745628509438465765378060600180801897, 749461441902427053579136913978468683133451, 2279376329849444925050273479606998902776609, 6932386594685332640776284412488747166013127, 21083830374489841315652151363876964507503481, 64123357402060461665162576283083648169972451, 195021724775751248000054346124916968715406977, 593129784144564254820679020118708706367952687, 1803916672585597573966366672817044701163229001, 5486346240942040573446128230083121950480657083, 16685912122734474876268164314378751931132941921, 50747738319886074280795466744368543910999407959, 154341754028223290707173873982692381057795449177, 469407658846816131149256530199673797854352709907
Generating function in Maple syntax:
(x^3+2*x-1)^2/(2*x^6-2*x^5+6*x^4-5*x^3+7*x^2-5*x+1)
Generating function in latex syntax:
\frac{\left(x^{3}+2 x -1\right)^{2}}{2 x^{6}-2 x^{5}+6 x^{4}-5 x^{3}+7 x^{2}-5 x +1}
Generating function in sympy syntax:
(x**3 + 2*x - 1)**2/(2*x**6 - 2*x**5 + 6*x**4 - 5*x**3 + 7*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^6-2*x^5+6*x^4-5*x^3+7*x^2-5*x+1)*F(x)-(x^3+2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{6}-2 x^{5}+6 x^{4}-5 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)-\left(x^{3}+2 x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-17024/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+4)-17024/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+4)-17024/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+4)-17024/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+4)-17024/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+4)-17024/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 6)^(-n+4)+7869/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+3)+7869/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+3)+7869/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+3)+7869/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+3)+7869/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+3)+7869/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 6)^(-n+3)-38411/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+2)-38411/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+2)-38411/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+2)-38411/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+2)-38411/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+2)-38411/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 6)^(-n+2)+23280/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+1)+23280/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+1)+23280/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+1)+23280/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+1)+23280/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+1)+23280/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 6)^(-n+1)+15042/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n-1)+15042/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n-1)+15042/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n-1)+15042/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n-1)+15042/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n-1)+15042/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 6)^(-n-1)-33597/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n)-33597/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n)-33597/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n)-33597/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n)-33597/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n)-33597/67913*RootOf(2*_Z^6-2*_Z^5+6*_Z^4-5*_Z^3+7*_Z^2-5*_Z+1,index = 6)^(-n)+piecewise(n = 0,1/2,0)
Explicit closed form in latex syntax:
-\frac{17024 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{67913}-\frac{17024 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{67913}-\frac{17024 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{67913}-\frac{17024 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{67913}-\frac{17024 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{67913}-\frac{17024 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{67913}+\frac{7869 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{67913}+\frac{7869 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{67913}+\frac{7869 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{67913}+\frac{7869 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{67913}+\frac{7869 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{67913}+\frac{7869 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{67913}-\frac{38411 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{67913}-\frac{38411 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{67913}-\frac{38411 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{67913}-\frac{38411 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{67913}-\frac{38411 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{67913}-\frac{38411 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{67913}+\frac{23280 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{67913}+\frac{23280 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{67913}+\frac{23280 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{67913}+\frac{23280 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{67913}+\frac{23280 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{67913}+\frac{23280 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{67913}+\frac{15042 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{67913}+\frac{15042 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{67913}+\frac{15042 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{67913}+\frac{15042 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{67913}+\frac{15042 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{67913}+\frac{15042 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{67913}-\frac{33597 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{67913}-\frac{33597 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{67913}-\frac{33597 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{67913}-\frac{33597 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{67913}-\frac{33597 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{67913}-\frac{33597 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+6 \textit{\_Z}^{4}-5 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{67913}+\left(\left\{\begin{array}{cc}\frac{1}{2} & 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) = 19
a(5) = 59
a(6) = 181
a(n+6) = -2*a(n)+2*a(n+1)-6*a(n+2)+5*a(n+3)-7*a(n+4)+5*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) = 19
a \! \left(5\right) = 59
a \! \left(6\right) = 181
a \! \left(n +6\right) = -2 a \! \left(n \right)+2 a \! \left(n +1\right)-6 a \! \left(n +2\right)+5 a \! \left(n +3\right)-7 a \! \left(n +4\right)+5 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19884/
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[23,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[21,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[11,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[20,x]+F[7,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[42,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[40,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = F[28,x]+F[32,x]
F[32,x] = 2*F[15,x]+F[33,x]+F[38,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[20,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[28,x]+F[37,x]
F[40,x] = F[41,x]+F[56,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[2,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[44,x]
F[47,x] = F[41,x]+F[48,x]
F[48,x] = 2*F[15,x]+F[49,x]+F[54,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[45,x]
F[52,x] = F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[41,x]+F[53,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[55,x]+F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[57,x]
F[61,x] = 3*F[15,x]+F[62,x]+F[67,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[53,x]
F[65,x] = F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[60,x]+F[66,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_{23}\! \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_{21}\! \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_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{42}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{40}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \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_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{45}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)
F_{61}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{67}\! \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_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{53}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{60}\! \left(x \right)+F_{66}\! \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_23(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_21(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_11(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_20(x) + F_7(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_42(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_40(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), F_28(x) + F_32(x))
Eq(F_32(x), 2*F_15(x) + F_33(x) + F_38(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_20(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_28(x) + F_37(x))
Eq(F_40(x), F_41(x) + F_56(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_2(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_44(x))
Eq(F_47(x), F_41(x) + F_48(x))
Eq(F_48(x), 2*F_15(x) + F_49(x) + F_54(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_45(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_41(x) + F_53(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_55(x) + F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_57(x))
Eq(F_61(x), 3*F_15(x) + F_62(x) + F_67(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_53(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_60(x) + F_66(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, 2, 0, 3], "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, 1, 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": [1, 2, 0, 3], "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, 1, 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": [1, 2, 0, 3], "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, 1, 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, 0], [0, 0], [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": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"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": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 0], [1, 0], [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": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], 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