0123_0132_0231_0312_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 140, 387, 1084, 3026, 8407, 23362, 65013, 180959, 503563, 1401142, 3898810, 10849228, 30190075, 84008944, 233768763, 650501724, 1810134146, 5037011059, 14016350211, 39002908937, 108532321268, 302009904931, 840394640559, 2338542999416, 6507399162507, 18107960363646, 50388522387685, 140214752852513, 390171729318873, 1085720120634369, 3021203464543626, 8407019636498961, 23393981900620005, 65097788851739607, 181145823374830135, 504069491528672881, 1402660285266136691, 3903144127798724100, 10861171619671884841, 30223083004240956794, 84100940328277115783, 234025369387621157694, 651215947208598384793, 1812120673106936927384, 5042538267033063381712, 14031732296777485192756, 39045714840806635723827, 108651434846695229343056, 302341354035299347370320, 841316955352463773921846, 2341109510546479336289311, 6514540929553757014467145, 18127833547147755053108926, 50443822928840783650038619, 140368636167039877850865675, 390599936233810054238675496, 1086911680215364394328720032, 3024519184461473023238839966, 8416246198921088155281435900, 23419656401837563918548140237, 65169232579061559309571918142, 181344627866128138083560713045, 504622699308423504145501602663, 1404199681312326655343228746568, 3907427762762009994136451538630, 10873091572656019092543262535829, 30256252328973388672247137953862, 84193239694282162111452595225742, 234282208290248161977850331816916, 651930645746167412814475271090340, 1814109445035078800934541942350777, 5048072367880136229443858924828097, 14047131886721536199664735283772505, 39088566855433449744846221860999140, 108770677967080095518174325369310438, 302673168580845114563710719063527773, 842240286545746786701606095800628045, 2343678838817146989702214062134755731, 6521690528538905356346594451396086797, 18147728539248219706141699276713518663, 50499184175184775245456295077976168598, 140522688381852697521838638692279418414, 391028613087312844484922768681387460414, 1088104547484116965632496354960153049582, 3027838543342724018365261978849041839926, 8425482887420439779218487823672519612004, 23445359080422863691235010851308394212750, 65240754714565598530511559367922263347131, 181543650541919750951380686518071896587839, 505176514224604473958475076017556659818636, 1405740766820119129252814624713278526286452, 3911716098942453223248887444708087285791633, 10885024607587248125525034178763195581458577, 30289458056481257364261142543378005279155842, 84285640357288791228324191049842389971167883
Generating function in Maple syntax:
-(2*x-1)/(x^8-x^7+8*x^6+2*x^5-2*x^4-x^3+x^2-3*x+1)
Generating function in latex syntax:
-\frac{2 x -1}{x^{8}-x^{7}+8 x^{6}+2 x^{5}-2 x^{4}-x^{3}+x^{2}-3 x +1}
Generating function in sympy syntax:
(1 - 2*x)/(x**8 - x**7 + 8*x**6 + 2*x**5 - 2*x**4 - x**3 + x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8-x^7+8*x^6+2*x^5-2*x^4-x^3+x^2-3*x+1)*F(x)+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}-x^{7}+8 x^{6}+2 x^{5}-2 x^{4}-x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)+2 x -1 = 0
Explicit closed form in Maple syntax:
-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+6)-541278243094/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n+6)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+5)+146750239164/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n+5)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+4)-4141840775632/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n+4)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+3)-3955529198943/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n+3)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+2)-1134309164195/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n+2)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+1)+1168263303096/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n+1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n-1)+1260825205499/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n-1)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 1)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 2)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 3)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 4)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 5)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 6)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 7)^(-n)+2901678975216/21315867328691*RootOf(_Z^8-_Z^7+8*_Z^6+2*_Z^5-2*_Z^4-_Z^3+_Z^2-3*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{21315867328691}-\frac{541278243094 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{21315867328691}+\frac{146750239164 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{21315867328691}-\frac{4141840775632 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{21315867328691}-\frac{3955529198943 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{21315867328691}-\frac{1134309164195 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{21315867328691}+\frac{1168263303096 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{21315867328691}+\frac{1260825205499 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{21315867328691}+\frac{2901678975216 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+8 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{21315867328691}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 140
a(7) = 387
a(n+8) = -a(n)+a(n+1)-8*a(n+2)-2*a(n+3)+2*a(n+4)+a(n+5)-a(n+6)+3*a(n+7), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 140
a \! \left(7\right) = 387
a \! \left(n +8\right) = -a \! \left(n \right)+a \! \left(n +1\right)-8 a \! \left(n +2\right)-2 a \! \left(n +3\right)+2 a \! \left(n +4\right)+a \! \left(n +5\right)-a \! \left(n +6\right)+3 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16050/
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[18,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[1,x]+F[12,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[20,x]+F[21,x]+F[69,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[20,x]+F[25,x]+F[27,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[4,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[12,x]+F[30,x]
F[33,x] = F[34,x]+F[47,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[2,x]*F[4,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[2,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[38,x]
F[47,x] = 2*F[20,x]+F[48,x]+F[50,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[38,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[68,x]
F[52,x] = F[53,x]+F[65,x]
F[53,x] = F[20,x]+F[54,x]+F[63,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[12,x]+F[57,x]
F[57,x] = F[20,x]+F[25,x]+F[31,x]
F[58,x] = F[41,x]+F[59,x]
F[59,x] = 2*F[20,x]+F[48,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[53,x]+F[62,x]
F[62,x] = F[60,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[2,x]+F[53,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[38,x]
F[68,x] = F[62,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[81,x]
F[71,x] = F[2,x]+F[72,x]
F[72,x] = F[20,x]+F[39,x]+F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[78,x]
F[75,x] = F[4,x]+F[76,x]
F[76,x] = F[20,x]+F[25,x]+F[77,x]
F[77,x] = F[12,x]*F[4,x]
F[78,x] = F[38,x]+F[79,x]
F[79,x] = 2*F[20,x]+F[48,x]+F[80,x]
F[80,x] = F[4,x]*F[53,x]
F[81,x] = F[53,x]+F[82,x]
F[82,x] = 2*F[20,x]+F[66,x]+F[83,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = F[15,x]
F[86,x] = F[44,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_{18}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{69}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{38}\! \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_{68}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{65}\! \left(x \right)
F_{53}\! \left(x \right) = F_{20}\! \left(x \right)+F_{54}\! \left(x \right)+F_{63}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{58}\! \left(x \right) = F_{41}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{53}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{2}\! \left(x \right)+F_{53}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{38}\! \left(x \right)
F_{68}\! \left(x \right) = F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{81}\! \left(x \right)
F_{71}\! \left(x \right) = F_{2}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{78}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{78}\! \left(x \right) = F_{38}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{81}\! \left(x \right) = F_{53}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{66}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{85}\! \left(x \right) = F_{15}\! \left(x \right)
F_{86}\! \left(x \right) = F_{44}\! \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_18(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_1(x) + F_12(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_69(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_20(x) + F_25(x) + F_27(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_4(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_12(x) + F_30(x))
Eq(F_33(x), F_34(x) + F_47(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_2(x)*F_4(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_2(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_38(x))
Eq(F_47(x), 2*F_20(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_38(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_68(x))
Eq(F_52(x), F_53(x) + F_65(x))
Eq(F_53(x), F_20(x) + F_54(x) + F_63(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_12(x) + F_57(x))
Eq(F_57(x), F_20(x) + F_25(x) + F_31(x))
Eq(F_58(x), F_41(x) + F_59(x))
Eq(F_59(x), 2*F_20(x) + F_48(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_53(x) + F_62(x))
Eq(F_62(x), F_60(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_2(x) + F_53(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_38(x))
Eq(F_68(x), F_62(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_81(x))
Eq(F_71(x), F_2(x) + F_72(x))
Eq(F_72(x), F_20(x) + F_39(x) + F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_78(x))
Eq(F_75(x), F_4(x) + F_76(x))
Eq(F_76(x), F_20(x) + F_25(x) + F_77(x))
Eq(F_77(x), F_12(x)*F_4(x))
Eq(F_78(x), F_38(x) + F_79(x))
Eq(F_79(x), 2*F_20(x) + F_48(x) + F_80(x))
Eq(F_80(x), F_4(x)*F_53(x))
Eq(F_81(x), F_53(x) + F_82(x))
Eq(F_82(x), 2*F_20(x) + F_66(x) + F_83(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), F_15(x))
Eq(F_86(x), F_44(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 3, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 3, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 3, 1, 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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [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, 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": [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": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [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", "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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, 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