0132_0231_0321_1203
Counting sequence:
1, 1, 2, 6, 20, 66, 213, 678, 2144, 6760, 21284, 66960, 210557, 661915, 2080515, 6539005, 20551551, 64592031, 203010216, 638060275, 2005439090, 6303185113, 19811277144, 62268152023, 195713183615, 615140731338, 1933432499300, 6076920877671, 19100210177769, 60033365836842, 188689277251031, 593064232429147, 1864044313732649, 5858827681546764, 18414723930533483, 57878823856782589, 181917375106060542, 571779609801586693, 1797145115182875924, 5648558482316025858, 17753832266029587861, 55801592773058638402, 175388485698192454478, 551258833131224323273, 1732646814973272517797, 5445835613233974131461, 17116659477774715747580, 53798912139420341351628, 169093914099376661911862, 531474534492681408445389, 1670463318084238721223456, 5250388336531091382998181, 16502354398286216718500915, 51868106363042245354238635, 163025250382818791655343925, 512400280749588888314087628, 1610511544044076415959854980, 5061955527628815854292504061, 15910096303531440296489701578, 50006595871104471558106092067, 157174386811278049511697981318, 494010588786666425251923316422, 1552711397731615919623430037786, 4880285441992425133119720704107, 15339093942453320601705479815487, 48211893704182600552199321517982, 151533506689746804845265170571462, 476280890159256828015675127469158, 1496985658725226781126886915504746, 4705135370179720643987579361757803, 14788584461502691405954206720898718, 46481602157738802106787251343145252, 146095073857445628692596340535718812, 459187498162838862123466601815225842, 1443259877980486500808637923385315642, 4536271313400539280140158364894264077, 14257832385373685072319285485760201911, 44813409579115519394062514649820280175, 140851822621067259964323300383934416393, 442707576192156386010233110603746233629, 1391462278377501477820716449351914640295, 4373467670919455133962120054260695204976, 13746128634461235195016237227039989266635, 43205087279273282766277011441651750880062, 135796748048021197924189444667544802245501, 426819107231946455871211600356106179980482, 1341523658827634327839552524807806431488833, 4216506938655581757037212469600218240297241, 13252789577537366325583048937625633423234120, 41654486555282201488692932757365753672783530, 130923096607906854787601143935103137558181289, 411500864442408567096184118108833286264260317, 1293377301821495145473408683111559670790137392, 4065179418599181235690896117314337463636738863, 12777156118426424629271269091084315302737868845, 40159535819675592421415690016116420078788929897, 126224357149862429359759671419358278378367950510, 396732382800353034132598460378360504060955605809, 1246958884294998375575538682820648431099448112929, 3919282938657165511857460022714809556957889059202, 12318592815459007652913065589779155839925780398239
Generating function in Maple syntax:
-(-1+x)^2*(2*x-1)^3/(3*x^6-26*x^5+56*x^4-58*x^3+32*x^2-9*x+1)
Generating function in latex syntax:
-\frac{\left(-1+x \right)^{2} \left(2 x -1\right)^{3}}{3 x^{6}-26 x^{5}+56 x^{4}-58 x^{3}+32 x^{2}-9 x +1}
Generating function in sympy syntax:
-(x - 1)**2*(2*x - 1)**3/(3*x**6 - 26*x**5 + 56*x**4 - 58*x**3 + 32*x**2 - 9*x + 1)
Implicit equation for the generating function in Maple syntax:
(3*x^6-26*x^5+56*x^4-58*x^3+32*x^2-9*x+1)*F(x)+(-1+x)^2*(2*x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{6}-26 x^{5}+56 x^{4}-58 x^{3}+32 x^{2}-9 x +1\right) F \! \left(x \right)+\left(-1+x \right)^{2} \left(2 x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
-2610/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 1)^(-n+4)-2610/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 2)^(-n+4)-2610/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 3)^(-n+4)-2610/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 4)^(-n+4)-2610/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 5)^(-n+4)-2610/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 6)^(-n+4)+26916/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 1)^(-n+3)+26916/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 2)^(-n+3)+26916/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 3)^(-n+3)+26916/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 4)^(-n+3)+26916/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 5)^(-n+3)+26916/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 6)^(-n+3)-80126/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 1)^(-n+2)-80126/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 2)^(-n+2)-80126/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 3)^(-n+2)-80126/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 4)^(-n+2)-80126/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 5)^(-n+2)-80126/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 6)^(-n+2)+12285/2221*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 1)^(-n+1)+12285/2221*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 2)^(-n+1)+12285/2221*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 3)^(-n+1)+12285/2221*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 4)^(-n+1)+12285/2221*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 5)^(-n+1)+12285/2221*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 6)^(-n+1)+7977/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 1)^(-n-1)+7977/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 2)^(-n-1)+7977/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 3)^(-n-1)+7977/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 4)^(-n-1)+7977/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 5)^(-n-1)+7977/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 6)^(-n-1)-41640/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 1)^(-n)-41640/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 2)^(-n)-41640/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 3)^(-n)-41640/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 4)^(-n)-41640/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 5)^(-n)-41640/15547*RootOf(3*_Z^6-26*_Z^5+56*_Z^4-58*_Z^3+32*_Z^2-9*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{2610 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{15547}-\frac{2610 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{15547}-\frac{2610 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{15547}-\frac{2610 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{15547}-\frac{2610 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{15547}-\frac{2610 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{15547}+\frac{26916 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{15547}+\frac{26916 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{15547}+\frac{26916 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{15547}+\frac{26916 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{15547}+\frac{26916 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{15547}+\frac{26916 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{15547}-\frac{80126 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{15547}-\frac{80126 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{15547}-\frac{80126 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{15547}-\frac{80126 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{15547}-\frac{80126 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{15547}-\frac{80126 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{15547}+\frac{12285 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{2221}+\frac{12285 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{2221}+\frac{12285 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{2221}+\frac{12285 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{2221}+\frac{12285 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{2221}+\frac{12285 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{2221}+\frac{7977 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{15547}+\frac{7977 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{15547}+\frac{7977 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{15547}+\frac{7977 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{15547}+\frac{7977 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{15547}+\frac{7977 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{15547}-\frac{41640 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{15547}-\frac{41640 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{15547}-\frac{41640 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{15547}-\frac{41640 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{15547}-\frac{41640 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{15547}-\frac{41640 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+56 \textit{\_Z}^{4}-58 \textit{\_Z}^{3}+32 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{15547}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(n+6) = -3*a(n)+26*a(n+1)-56*a(n+2)+58*a(n+3)-32*a(n+4)+9*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) = 66
a \! \left(n +6\right) = -3 a \! \left(n \right)+26 a \! \left(n +1\right)-56 a \! \left(n +2\right)+58 a \! \left(n +3\right)-32 a \! \left(n +4\right)+9 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15311/
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[12,x]*F[4,x]
F[4,x] = F[18,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[15,x]+F[20,x]+F[58,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[23,x]+F[6,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[19,x]+F[30,x]
F[30,x] = 2*F[15,x]+F[31,x]+F[35,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[23,x]
F[34,x] = F[30,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]+F[52,x]
F[38,x] = F[15,x]+F[39,x]+F[51,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[12,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = x^2
F[44,x] = F[38,x]+F[45,x]
F[45,x] = 2*F[15,x]+F[46,x]+F[50,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[42,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[38,x]
F[51,x] = F[12,x]*F[2,x]
F[52,x] = 2*F[15,x]+F[53,x]+F[57,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[27,x]
F[56,x] = F[52,x]
F[57,x] = F[12,x]*F[37,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[73,x]
F[60,x] = F[2,x]+F[61,x]
F[61,x] = F[15,x]+F[62,x]+F[72,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[10,x]+F[65,x]
F[65,x] = F[28,x]
F[66,x] = F[61,x]+F[67,x]
F[67,x] = 2*F[15,x]+F[57,x]+F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[65,x]
F[71,x] = F[67,x]
F[72,x] = F[12,x]*F[60,x]
F[73,x] = F[38,x]+F[74,x]
F[74,x] = F[15,x]+F[53,x]+F[75,x]+F[76,x]
F[75,x] = F[12,x]*F[61,x]
F[76,x] = F[12,x]*F[73,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{58}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = x^{2}
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{27}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{73}\! \left(x \right)
F_{60}\! \left(x \right) = F_{2}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{72}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{28}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{65}\! \left(x \right)
F_{71}\! \left(x \right) = F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{73}\! \left(x \right) = F_{38}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_18(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_58(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_23(x) + F_6(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_19(x) + F_30(x))
Eq(F_30(x), 2*F_15(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_23(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x) + F_52(x))
Eq(F_38(x), F_15(x) + F_39(x) + F_51(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_12(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), x**2)
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), 2*F_15(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_42(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_38(x))
Eq(F_51(x), F_12(x)*F_2(x))
Eq(F_52(x), 2*F_15(x) + F_53(x) + F_57(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_27(x))
Eq(F_56(x), F_52(x))
Eq(F_57(x), F_12(x)*F_37(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_73(x))
Eq(F_60(x), F_2(x) + F_61(x))
Eq(F_61(x), F_15(x) + F_62(x) + F_72(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_10(x) + F_65(x))
Eq(F_65(x), F_28(x))
Eq(F_66(x), F_61(x) + F_67(x))
Eq(F_67(x), 2*F_15(x) + F_57(x) + F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_65(x))
Eq(F_71(x), F_67(x))
Eq(F_72(x), F_12(x)*F_60(x))
Eq(F_73(x), F_38(x) + F_74(x))
Eq(F_74(x), F_15(x) + F_53(x) + F_75(x) + F_76(x))
Eq(F_75(x), F_12(x)*F_61(x))
Eq(F_76(x), F_12(x)*F_73(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]]}], "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]]}], "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]]}], "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, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 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