0231_0312_1032_2103
Counting sequence:
1, 1, 2, 6, 20, 63, 187, 543, 1579, 4623, 13588, 39964, 117471, 345123, 1013802, 2978189, 8749464, 25705418, 75521245, 221876350, 651855190, 1915097670, 5626405634, 16529946573, 48563723596, 142676521093, 419172729978, 1231497455591, 3618045439717, 10629541192011, 31228780465866, 91747772949381, 269547952774333, 791911307323204, 2326574964297588, 6835299628878364, 20081588489399285, 58998173918530891, 173332130954634339, 509236568286280261, 1496098161672998383, 4395422184422886292, 12913414824171126932, 37938626922441839471, 111460789602113255561, 327463290754548520634, 962062149161208909455, 2826465148860040330000, 8303938830444741819260, 24396338347770563299623, 71674579609956272278872, 210574443141116148317050, 618651638356202327407216, 1817551284627303645948856, 5339826919440420015226653, 15688003838322210988945396, 46090157629491814121691781, 135409364518525765587299228, 397822375586289735394914553, 1168771769071160564197485547, 3433762231610403607523663897, 10088131297528099776480097072, 29638159607934351853637579321, 87074650303236159881161082257, 255818675171757620380270023342, 751575738044635246382652963782, 2208072141871913633575309730052, 6487147385033422773107585910927, 19058743777940902265653574086883, 55993134244371666995138653929427, 164503553803846559052246611809283, 483298882609258594330841075866331, 1419895221290326972603602006796956, 4171543763061214186269429950364096, 12255677113499321968561603202993891, 36006243741316583547289887583521359, 105783595337306400752153556827038282, 310784127410837196679133383357652305, 913060040571834340016016234381793836, 2682500694724890842172673429614968762, 7880982254675065754338362462781886141, 23153724217347531590088113840901841814, 68023874157941350486267861140553601834, 199848949223838259020788437811757162378, 587140956028738486016247145140902062542, 1724975305525502418655770842892896815465, 5067845760238804053998078294280467127936, 14888943956067965717899734124354872782429, 43742580696948223590513371690045442706618, 128512362708519614450364333565464052579147, 377559510796726496361431712264304275534689, 1109241019219181517382333116169233670118435, 3258865433218684338531510986642754711589702, 9574297855756715412550280540025210911737845, 28128556182883162277944722689633345010218517, 82639550685994399214399683854060259769571352, 242788691078954560907526373258926655758191196, 713294639509967107700590606646510900165525196, 2095605196817821885650014722818751722514436205, 6156728086372359778108336289238796488098344107, 18087997103216525573792647463053272253880591159
Generating function in Maple syntax:
(2*x-1)*(x-1)^3/(2*x^6+x^5+6*x^4-14*x^3+13*x^2-6*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{2 x^{6}+x^{5}+6 x^{4}-14 x^{3}+13 x^{2}-6 x +1}
Generating function in sympy syntax:
(x - 1)**3*(2*x - 1)/(2*x**6 + x**5 + 6*x**4 - 14*x**3 + 13*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^6+x^5+6*x^4-14*x^3+13*x^2-6*x+1)*F(x)-(2*x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{6}+x^{5}+6 x^{4}-14 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
-8917680/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+4)-8917680/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+4)-8917680/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+4)-8917680/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+4)-8917680/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+4)-8917680/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+4)-9716004/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+3)-9716004/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+3)-9716004/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+3)-9716004/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+3)-9716004/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+3)-9716004/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+3)-32646896/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+2)-32646896/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+2)-32646896/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+2)-32646896/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+2)-32646896/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+2)-32646896/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+2)+43045833/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+1)+43045833/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+1)+43045833/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+1)+43045833/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+1)+43045833/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+1)+43045833/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+1)+9464610/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n-1)+9464610/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n-1)+9464610/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n-1)+9464610/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n-1)+9464610/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n-1)+9464610/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n-1)-25985977/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n)-25985977/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n)-25985977/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n)-25985977/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n)-25985977/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n)-25985977/42909577*RootOf(2*_Z^6+_Z^5+6*_Z^4-14*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{8917680 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{42909577}-\frac{9716004 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{42909577}-\frac{32646896 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{42909577}+\frac{43045833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{42909577}+\frac{9464610 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{42909577}-\frac{25985977 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}+\textit{\_Z}^{5}+6 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{42909577}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(n+6) = -2*a(n)-a(n+1)-6*a(n+2)+14*a(n+3)-13*a(n+4)+6*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) = 63
a \! \left(n +6\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)-6 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15775/
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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[61,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[32,x]
F[25,x] = F[17,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[29,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[25,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[11,x]+F[31,x]
F[31,x] = F[15,x]
F[32,x] = F[33,x]+F[36,x]
F[33,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] = 2*F[22,x]+F[37,x]+F[39,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[32,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[58,x]
F[41,x] = F[22,x]+F[42,x]+F[56,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[4,x]+F[45,x]
F[45,x] = F[22,x]+F[46,x]+F[48,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[44,x]
F[48,x] = F[11,x]*F[4,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[51,x]
F[51,x] = F[2,x]*F[4,x]
F[52,x] = 2*F[22,x]+F[53,x]+F[55,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[49,x]
F[55,x] = F[4,x]*F[41,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[2,x]+F[41,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[33,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[20,x]+F[63,x]
F[63,x] = F[41,x]+F[64,x]
F[64,x] = 2*F[22,x]+F[59,x]+F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[73,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[69,x]
F[69,x] = F[17,x]*F[4,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[67,x]
F[73,x] = F[74,x]+F[76,x]
F[74,x] = F[75,x]
F[75,x] = F[33,x]*F[4,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,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_{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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \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_{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_{1}\! \left(x \right)+F_{17}\! \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_{61}\! \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_{32}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)+F_{31}\! \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_{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) = 2 F_{22}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{58}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \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_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{52}\! \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) = 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \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_{33}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{41}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \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_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_6(x))
Eq(F_10(x), F_11(x) + F_14(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_11(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))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_61(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_32(x))
Eq(F_25(x), F_17(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_29(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_11(x) + F_31(x))
Eq(F_31(x), F_15(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(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), 2*F_22(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_32(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_58(x))
Eq(F_41(x), F_22(x) + F_42(x) + F_56(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_4(x) + F_45(x))
Eq(F_45(x), F_22(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_11(x)*F_4(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_2(x)*F_4(x))
Eq(F_52(x), 2*F_22(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_49(x))
Eq(F_55(x), F_4(x)*F_41(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_2(x) + F_41(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_33(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_20(x) + F_63(x))
Eq(F_63(x), F_41(x) + F_64(x))
Eq(F_64(x), 2*F_22(x) + F_59(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_73(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_17(x)*F_4(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_67(x))
Eq(F_73(x), F_74(x) + F_76(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_33(x)*F_4(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 0], [0, 1]]}, {"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", 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