0132_0231_1023_2031
Counting sequence:
1, 1, 2, 6, 20, 65, 205, 638, 1979, 6138, 19043, 59086, 183323, 568751, 1764444, 5473720, 16980540, 52676545, 163410983, 506925440, 1572556437, 4878294008, 15133153767, 46945153606, 145630382845, 451765604265, 1401439324914, 4347458220194, 13486415079744, 41836719160117, 129783270189745, 402605592777422, 1248938037209387, 3874377915452902, 12018854229966975, 37284142134782346, 115660546957834791, 358794955514024815, 1113031396317724948, 3452776774079692596, 10710989366779957952, 33226965053762907309, 103074624467662727851, 319751689382645314680, 991913804102736380349, 3077053311787395318996, 9545443408882785557307, 29611280871531545604314, 91858273868742553295545, 284957024140466831492353, 883975957603837081323602, 2742215237467657130998910, 8506729559684163667227196, 26389047370484282100048601, 81862461506003341006214197, 253948636710438278067993166, 787783666660934126563250251, 2443813495109931371466160754, 7581046233409469508422052235, 23517450127881526757361359574, 72954371136799899717976021747, 226314512798984945498324567327, 702058806146108749580029567804, 2177883164413262909656675610160, 6756093700856736390408430496852, 20958333688691282443238858226505, 65015639281443153831916971405903, 201687472580683282072408266700960, 625662333640918125279014581343445, 1940890778828130237734407356372624, 6020910661855610111299093482689663, 18677694589251635354849471066607294, 57940782509774744200519875035011253, 179740291918946629116799943086064009, 557579154780278592652387211981698338, 1729687375748153695717438157119354010, 5365728600455840331420126236434546952, 16645229546926999447810853517203943085, 51635795863091906089560064427039034073, 160181354477453667391654116720509606398, 496904635502509394012360072043217642907, 1541466655650212641893037421905038660254, 4781841988812462410511498451646034619911, 14833932814670399234523620825315389766578, 46016903792507388142669183717879021104895, 142750776958802857251405123050349888237887, 442832582005655351833102685224026224148180, 1373727694262491671246104328804381964341932, 4261492615192532405988599463945115721732344, 13219737350559970677108020418248019353331477, 41009447017402541139406465607174288495716243, 127216955986036168287615118434693002512923128, 394644528698113047586053745269094301844037469, 1224241712311139389656621360323147590068135724, 3797766499149939404411577884328781120194972723, 11781195034465540953959179258336515545469707506, 36546890513459052899329953380130725120909865073, 113373490744808538260647450193160178858015785665, 351700191810562811007765920358594203100913278562, 1091022461308934107416350666704219389326427112438, 3384502024160837452745664519242786377875509566884
Generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^3-2*x^2+3*x-1)/(x^6-5*x^5+16*x^4-23*x^3+18*x^2-7*x+1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}{x^{6}-5 x^{5}+16 x^{4}-23 x^{3}+18 x^{2}-7 x +1}
Generating function in sympy syntax:
(1 - x)*(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1)/(x**6 - 5*x**5 + 16*x**4 - 23*x**3 + 18*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6-5*x^5+16*x^4-23*x^3+18*x^2-7*x+1)*F(x)+(x-1)*(2*x-1)*(x^3-2*x^2+3*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}-5 x^{5}+16 x^{4}-23 x^{3}+18 x^{2}-7 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) = 0
Explicit closed form in Maple syntax:
-5395/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+4)-5395/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+4)-5395/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+4)-5395/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+4)-5395/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+4)-5395/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+4)+24215/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+3)+24215/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+3)+24215/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+3)+24215/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+3)+24215/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+3)+24215/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+3)-73523/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+2)-73523/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+2)-73523/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+2)-73523/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+2)-73523/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+2)-73523/18778*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+2)+41439/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+1)+41439/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+1)+41439/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+1)+41439/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+1)+41439/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+1)+41439/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+1)+4652/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n-1)+4652/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n-1)+4652/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n-1)+4652/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n-1)+4652/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n-1)+4652/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n-1)-22178/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n)-22178/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n)-22178/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n)-22178/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n)-22178/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n)-22178/9389*RootOf(_Z^6-5*_Z^5+16*_Z^4-23*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{5395 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{18778}+\frac{24215 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{18778}-\frac{73523 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{18778}+\frac{41439 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{9389}+\frac{4652 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{9389}-\frac{22178 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-5 \textit{\_Z}^{5}+16 \textit{\_Z}^{4}-23 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{9389}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+6) = -a(n)+5*a(n+1)-16*a(n+2)+23*a(n+3)-18*a(n+4)+7*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) = 20
a \! \left(5\right) = 65
a \! \left(n +6\right) = -a \! \left(n \right)+5 a \! \left(n +1\right)-16 a \! \left(n +2\right)+23 a \! \left(n +3\right)-18 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15323/
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[22,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[50,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[1,x]+F[28,x]
F[31,x] = F[15,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[24,x]+F[25,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[2,x]+F[33,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[40,x]+F[47,x]
F[40,x] = F[24,x]+F[41,x]+F[46,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[21,x]+F[4,x]
F[44,x] = F[40,x]+F[45,x]
F[45,x] = F[37,x]
F[46,x] = F[2,x]*F[4,x]
F[47,x] = F[48,x]
F[48,x] = F[39,x]*F[4,x]
F[49,x] = F[45,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[2,x]+F[53,x]
F[53,x] = F[24,x]+F[41,x]+F[54,x]
F[54,x] = F[4,x]*F[52,x]
F[55,x] = F[23,x]+F[56,x]
F[56,x] = F[37,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_{22}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{50}\! \left(x \right)
F_{24}\! \left(x \right) = 0
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_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \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_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{40}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{2}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{37}\! \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_22(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_50(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_32(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_1(x) + F_28(x))
Eq(F_31(x), F_15(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_24(x) + F_25(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_2(x) + F_33(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_40(x) + F_47(x))
Eq(F_40(x), F_24(x) + F_41(x) + F_46(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_21(x) + F_4(x))
Eq(F_44(x), F_40(x) + F_45(x))
Eq(F_45(x), F_37(x))
Eq(F_46(x), F_2(x)*F_4(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_39(x)*F_4(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_2(x) + F_53(x))
Eq(F_53(x), F_24(x) + F_41(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_52(x))
Eq(F_55(x), F_23(x) + F_56(x))
Eq(F_56(x), F_37(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": [1, 0, 2, 3], "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, 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": [1, 0, 2, 3], "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, 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": [1, 0, 2, 3], "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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 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, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 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, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 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, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 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, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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], "pos": [[0, 0]]}], "requirements": []}, {"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": [1, 0, 2, 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, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": 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