0123_0213_0321_1032
Counting sequence:
1, 1, 2, 6, 20, 64, 206, 665, 2145, 6919, 22320, 72001, 232264, 749249, 2416965, 7796766, 25151197, 81133987, 261726065, 844289005, 2723549617, 8785762306, 28341550606, 91425588672, 294925227635, 951384521106, 3069023678505, 9900209778770, 31936590893752, 103022649085887, 332335604009326, 1072064780630588, 3458320083683416, 11156021555127682, 35987651208362406, 116090761665790403, 374491373891378097, 1208052967408335557, 3896997564721969652, 12571129270953334759, 40552576316093892348, 130816524945965124483, 421994476147420060325, 1361290845881226852944, 4391319962284666177933, 14165739136133375131049, 45696548417432871938981, 147410206922444195520371, 475523203775015959538069, 1533966487459198517316108, 4948345666347777728453542, 15962620457387359297803042, 51493017878572208654342331, 166108747452919117918235032, 535841889194448860231669069, 1728545513816829289545112008, 5576028402385753027352902916, 17987430759377380067619442849, 58024752023315158142375984102, 187179141502016155812178202006, 603811818090214652724455459664, 1947806303308017843175710333592, 6283330801981085927651560735674, 20269082146450492049436072739153, 65385016961071785214244062314805, 210922251541040561583523270282247, 680403527640420774390850452133188, 2194879663208268676873235687532081, 7080352379523803059014137188077700, 22840154136264139884998480307329017, 73678909326175312351619203018002113, 237677103539139768316570025972893030, 766710664739523697712286394012377553, 2473293534261360481744846594788948883, 7978473742369626322421997319061443813, 25737358860112906383884712606968349021, 83024856944317577889112485614229136493, 267825727887999751902797185584320950122, 863965602094859198517489413541732632250, 2787023365863043155970291425659159659304, 8990519093622123941518618865182922111479, 29002065272514837009918845752398353981922, 93556309853999604522743320221474846522685, 301798614383251099328942437223070665633762, 973557035177958701935029121782499481991844, 3140548881184982202197509343081467117920439, 10130939347903078571592350081605480735505754, 32680889855204094784914333889965400967243348, 105423645828957313952942728135354130444984988, 340080859153827962069881273527492828639997586, 1097049811296112054719267092967381227868719450, 3538918042783614934244603286185193805888706179, 11416018474806510982465975872785746161960635565, 36826362250145009092631052016314431111240826525, 118796317610364690657235488993428758165206174224, 383219091310792547088545306752745632541083019423, 1236207273921903534523877857468641317405181547264, 3987819132053731042423833332067396170682792805235, 12864106016398034876240198962600782147179254489961, 41497675326088085129022508794686464697559901246304, 133865272508971276754634035176330871211752512885865
Generating function in Maple syntax:
(x^2+x+1)*(x^4-x^3+x^2-3*x+1)/(x^6+2*x^4-3*x^3-3*x+1)
Generating function in latex syntax:
\frac{\left(x^{2}+x +1\right) \left(x^{4}-x^{3}+x^{2}-3 x +1\right)}{x^{6}+2 x^{4}-3 x^{3}-3 x +1}
Generating function in sympy syntax:
(x**2 + x + 1)*(x**4 - x**3 + x**2 - 3*x + 1)/(x**6 + 2*x**4 - 3*x**3 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6+2*x^4-3*x^3-3*x+1)*F(x)-(x^2+x+1)*(x^4-x^3+x^2-3*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+2 x^{4}-3 x^{3}-3 x +1\right) F \! \left(x \right)-\left(x^{2}+x +1\right) \left(x^{4}-x^{3}+x^{2}-3 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,0)-4715/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 6)^(-n+4)-4715/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 1)^(-n+4)-4715/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 2)^(-n+4)-4715/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 3)^(-n+4)-4715/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 4)^(-n+4)-4715/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 5)^(-n+4)+421/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 1)^(-n+3)+421/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 2)^(-n+3)+421/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 3)^(-n+3)+421/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 4)^(-n+3)+421/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 5)^(-n+3)+421/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 6)^(-n+3)-2713/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 1)^(-n+2)-2713/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 2)^(-n+2)-2713/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 3)^(-n+2)-2713/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 4)^(-n+2)-2713/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 5)^(-n+2)-2713/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 6)^(-n+2)+21019/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 1)^(-n+1)+21019/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 2)^(-n+1)+21019/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 3)^(-n+1)+21019/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 4)^(-n+1)+21019/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 5)^(-n+1)+21019/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 6)^(-n+1)+2586/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 1)^(-n-1)+2586/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 2)^(-n-1)+2586/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 3)^(-n-1)+2586/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 4)^(-n-1)+2586/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 5)^(-n-1)+2586/64373*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 6)^(-n-1)+629/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 1)^(-n)+629/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 2)^(-n)+629/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 3)^(-n)+629/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 4)^(-n)+629/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 5)^(-n)+629/128746*RootOf(_Z^6+2*_Z^4-3*_Z^3-3*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{4715 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{128746}-\frac{4715 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{128746}-\frac{4715 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{128746}-\frac{4715 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{128746}-\frac{4715 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{128746}-\frac{4715 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{128746}+\frac{421 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{64373}+\frac{421 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{64373}+\frac{421 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{64373}+\frac{421 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{64373}+\frac{421 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{64373}+\frac{421 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{64373}-\frac{2713 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{128746}-\frac{2713 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{128746}-\frac{2713 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{128746}-\frac{2713 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{128746}-\frac{2713 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{128746}-\frac{2713 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{128746}+\frac{21019 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{128746}+\frac{21019 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{128746}+\frac{21019 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{128746}+\frac{21019 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{128746}+\frac{21019 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{128746}+\frac{21019 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{128746}+\frac{2586 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{64373}+\frac{2586 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{64373}+\frac{2586 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{64373}+\frac{2586 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{64373}+\frac{2586 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{64373}+\frac{2586 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{64373}+\frac{629 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{128746}+\frac{629 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{128746}+\frac{629 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{128746}+\frac{629 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{128746}+\frac{629 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{128746}+\frac{629 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{128746}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 206
a(n) = -2*a(n+2)+3*a(n+3)+3*a(n+5)-a(n+6), n >= 7
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(6\right) = 206
a \! \left(n \right) = -2 a \! \left(n +2\right)+3 a \! \left(n +3\right)+3 a \! \left(n +5\right)-a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14907/
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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[84,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[37,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[35,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[15,x]*F[4,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[15,x]+F[33,x]
F[37,x] = F[38,x]+F[73,x]
F[38,x] = F[22,x]+F[23,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[22,x]+F[42,x]+F[72,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[4,x]+F[45,x]
F[45,x] = F[22,x]+F[27,x]+F[34,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[22,x]+F[48,x]+F[65,x]+F[66,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[57,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[4,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[33,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = 2*F[22,x]+F[48,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[41,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[38,x]*F[4,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[71,x]
F[68,x] = F[41,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[41,x]
F[71,x] = F[64,x]
F[72,x] = F[2,x]*F[4,x]
F[73,x] = F[22,x]+F[66,x]+F[74,x]+F[82,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[79,x]
F[76,x] = F[77,x]+F[78,x]
F[77,x] = F[52,x]
F[78,x] = F[55,x]
F[79,x] = F[80,x]+F[81,x]
F[80,x] = 2*F[22,x]+F[59,x]+F[74,x]
F[81,x] = F[62,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[38,x]+F[64,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[89,x]
F[86,x] = F[2,x]+F[87,x]
F[87,x] = F[22,x]+F[42,x]+F[88,x]
F[88,x] = F[4,x]*F[40,x]
F[89,x] = F[38,x]+F[90,x]
F[90,x] = F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[38,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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \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_{15}\! \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_{84}\! \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_{37}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{73}\! \left(x \right)
F_{38}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \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_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{72}\! \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_{46}\! \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_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \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_{57}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{33}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{48}\! \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_{41}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \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_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \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_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{41}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{71}\! \left(x \right) = F_{64}\! \left(x \right)
F_{72}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{22}\! \left(x \right)+F_{66}\! \left(x \right)+F_{74}\! \left(x \right)+F_{82}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{52}\! \left(x \right)
F_{78}\! \left(x \right) = F_{55}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{81}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{74}\! \left(x \right)
F_{81}\! \left(x \right) = F_{62}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{38}\! \left(x \right)+F_{64}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{89}\! \left(x \right)
F_{86}\! \left(x \right) = F_{2}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{89}\! \left(x \right) = F_{38}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{38}\! \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_14(x))
Eq(F_10(x), F_1(x) + F_11(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_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_84(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_37(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_35(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_15(x)*F_4(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_15(x) + F_33(x))
Eq(F_37(x), F_38(x) + F_73(x))
Eq(F_38(x), F_22(x) + F_23(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_22(x) + F_42(x) + F_72(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_4(x) + F_45(x))
Eq(F_45(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_22(x) + F_48(x) + F_65(x) + F_66(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_57(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_4(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_33(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), 2*F_22(x) + F_48(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_41(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_38(x)*F_4(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_71(x))
Eq(F_68(x), F_41(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_41(x))
Eq(F_71(x), F_64(x))
Eq(F_72(x), F_2(x)*F_4(x))
Eq(F_73(x), F_22(x) + F_66(x) + F_74(x) + F_82(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_79(x))
Eq(F_76(x), F_77(x) + F_78(x))
Eq(F_77(x), F_52(x))
Eq(F_78(x), F_55(x))
Eq(F_79(x), F_80(x) + F_81(x))
Eq(F_80(x), 2*F_22(x) + F_59(x) + F_74(x))
Eq(F_81(x), F_62(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_38(x) + F_64(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_89(x))
Eq(F_86(x), F_2(x) + F_87(x))
Eq(F_87(x), F_22(x) + F_42(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_40(x))
Eq(F_89(x), F_38(x) + F_90(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_38(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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, 0], [0, 0], [0, 1], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", 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