0132_0213_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 64, 197, 596, 1796, 5416, 16352, 49400, 149261, 450975, 1362499, 4116301, 12435795, 37569825, 113502360, 342902523, 1035944290, 3129694937, 9455130320, 28564917303, 86297534855, 260713662850, 787642573676, 2379548538543, 7188858738393, 21718275058952, 65613122620279, 198223930725797, 598854696925461, 1809201070891138, 5465780814842247, 16512680866564775, 49886491717912762, 150712175433784959, 455316841097308544, 1375558578339225148, 4155702648366180445, 12554801208259143686, 37929333909235921282, 114588383115405915599, 346183183083999130413, 1045854675593107201229, 3159633557916231329408, 9545574976409058709318, 28838154792320851819246, 87123004521038114714825, 263207475354579985039860, 795176606493116973124995, 2402309564582283614382091, 7257622013724591137433065, 21926015726976967343244633, 66240727989210004882538148, 200119989841192803635345840, 604582883517860842437044357, 1826506504087034214534371130, 5518062287937190550398383253, 16670628517019319848504072838, 50363667651955441647751782036, 152153772532755308599176487014, 459672052796785821439449380207, 1388716116630780107573993597063, 4195452912258404647067126048844, 12674890806108034281464270187998, 38292136822072644274043840791732, 115684447687212189105138048554986, 349494505853250411052025086563761, 1055858519131866016230657980715770, 3189856188730609548349732411620444, 9636880624071749354460642820390156, 29113998458835393679653534711336640, 87956356348734497435314527547949186, 265725116152768817442427446227286647, 802782655916810302976411335513361159, 2425288215022698235840612238432831323, 7327042858456473961172971759034624557, 22135743173582185732931355582729250459, 66874335978705140879617845163391415571, 202034184148379460097720974938408451760, 610365859595210893744316184674955058615, 1843977464159195543243492682673626247002, 5570843838778915620862377733009516250369, 16830086961074557894589011562112180425416, 50845407861839900453904944388497678617505, 153609158800911437674880521510528753660901, 464068923031150352678228689335771789924556, 1401999509693388901393835568645161354302794, 4235583396409767700570107090774907214857681, 12796129088422818743618763318634192641388311, 38658410028326504094048958779043804616464220, 116790996370169925306696181412707715999156297, 352837502192728353759441748476154652219913345, 1065958051757842474842606742788342228774435971, 3220367906036017031547594319769430865261337162, 9729059631497362171437221921037399888379118993, 29392480634221349337721952009820760654964859477, 88797679380665361199426159792952206600385216674, 268266838771375732063795174547957536226753531379
Generating function in Maple syntax:
-(2*x-1)^2*(x-1)^2/(x^6-12*x^4+22*x^3-18*x^2+7*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right)^{2} \left(x -1\right)^{2}}{x^{6}-12 x^{4}+22 x^{3}-18 x^{2}+7 x -1}
Generating function in sympy syntax:
-(x - 1)**2*(2*x - 1)**2/(x**6 - 12*x**4 + 22*x**3 - 18*x**2 + 7*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^6-12*x^4+22*x^3-18*x^2+7*x-1)*F(x)+(2*x-1)^2*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}-12 x^{4}+22 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
139578/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 1)^(-n+4)+139578/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 2)^(-n+4)+139578/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 3)^(-n+4)+139578/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 4)^(-n+4)+139578/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 5)^(-n+4)+139578/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 6)^(-n+4)+88570/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 1)^(-n+3)+88570/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 2)^(-n+3)+88570/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 3)^(-n+3)+88570/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 4)^(-n+3)+88570/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 5)^(-n+3)+88570/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 6)^(-n+3)-1622880/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 1)^(-n+2)-1622880/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 2)^(-n+2)-1622880/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 3)^(-n+2)-1622880/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 4)^(-n+2)-1622880/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 5)^(-n+2)-1622880/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 6)^(-n+2)+2026885/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 1)^(-n+1)+2026885/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 2)^(-n+1)+2026885/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 3)^(-n+1)+2026885/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 4)^(-n+1)+2026885/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 5)^(-n+1)+2026885/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 6)^(-n+1)+247945/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 1)^(-n-1)+247945/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 2)^(-n-1)+247945/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 3)^(-n-1)+247945/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 4)^(-n-1)+247945/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 5)^(-n-1)+247945/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 6)^(-n-1)-1109172/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 1)^(-n)-1109172/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 2)^(-n)-1109172/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 3)^(-n)-1109172/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 4)^(-n)-1109172/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 5)^(-n)-1109172/533923*RootOf(_Z^6-12*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{139578 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{533923}+\frac{139578 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{533923}+\frac{139578 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{533923}+\frac{139578 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{533923}+\frac{139578 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{533923}+\frac{139578 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{533923}+\frac{88570 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{533923}+\frac{88570 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{533923}+\frac{88570 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{533923}+\frac{88570 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{533923}+\frac{88570 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{533923}+\frac{88570 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{533923}-\frac{1622880 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{533923}-\frac{1622880 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{533923}-\frac{1622880 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{533923}-\frac{1622880 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{533923}-\frac{1622880 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{533923}-\frac{1622880 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{533923}+\frac{2026885 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{533923}+\frac{2026885 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{533923}+\frac{2026885 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{533923}+\frac{2026885 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{533923}+\frac{2026885 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{533923}+\frac{2026885 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{533923}+\frac{247945 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{533923}+\frac{247945 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{533923}+\frac{247945 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{533923}+\frac{247945 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{533923}+\frac{247945 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{533923}+\frac{247945 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{533923}-\frac{1109172 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{533923}-\frac{1109172 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{533923}-\frac{1109172 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{533923}-\frac{1109172 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{533923}-\frac{1109172 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{533923}-\frac{1109172 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-12 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{533923}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n) = 12*a(n+2)-22*a(n+3)+18*a(n+4)-7*a(n+5)+a(n+6), 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) = 64
a \! \left(n \right) = 12 a \! \left(n +2\right)-22 a \! \left(n +3\right)+18 a \! \left(n +4\right)-7 a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15285/
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[16,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[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[62,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[37,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[18,x]+F[23,x]+F[31,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[13,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[31,x]
F[35,x] = F[36,x]
F[36,x] = F[33,x]*F[4,x]
F[37,x] = F[38,x]+F[47,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[2,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[44,x]
F[47,x] = 2*F[18,x]+F[48,x]+F[56,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[42,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[50,x]
F[53,x] = F[45,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[53,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[44,x]+F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[56,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[58,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[73,x]
F[64,x] = F[2,x]+F[65,x]
F[65,x] = F[18,x]+F[66,x]+F[72,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[11,x]+F[69,x]
F[69,x] = F[15,x]+F[18,x]+F[23,x]
F[70,x] = F[42,x]+F[71,x]
F[71,x] = 2*F[18,x]+F[46,x]+F[48,x]
F[72,x] = F[4,x]*F[64,x]
F[73,x] = F[38,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[4,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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{62}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
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_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \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_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \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) = 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{42}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{45}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{44}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \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_{73}\! \left(x \right)
F_{64}\! \left(x \right) = F_{2}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{72}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{11}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{70}\! \left(x \right) = F_{42}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{73}\! \left(x \right) = F_{38}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \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_16(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_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_62(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_37(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_31(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_13(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_31(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_33(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_47(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_2(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_38(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), 2*F_18(x) + F_48(x) + F_56(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_42(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_50(x))
Eq(F_53(x), F_45(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_53(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_44(x) + F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_58(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_73(x))
Eq(F_64(x), F_2(x) + F_65(x))
Eq(F_65(x), F_18(x) + F_66(x) + F_72(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_11(x) + F_69(x))
Eq(F_69(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_70(x), F_42(x) + F_71(x))
Eq(F_71(x), 2*F_18(x) + F_46(x) + F_48(x))
Eq(F_72(x), F_4(x)*F_64(x))
Eq(F_73(x), F_38(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(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, 1, 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, 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": [0, 2, 1, 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, 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": [0, 2, 1, 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, 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, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [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": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [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, 1, 0, 3], "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, 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, 2], [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": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [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, 1, 0, 3], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"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, 1], [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": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], 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