0123_0132_1302_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 50, 111, 231, 470, 944, 1880, 3719, 7314, 14307, 27845, 53936, 104009, 199735, 382085, 728313, 1383738, 2621120, 4951418, 9330109, 17541038, 32909755, 61627815, 115209066, 215041753, 400818371, 746136015, 1387348899, 2576901106, 4781861104, 8865886824, 16425127219, 30408059318, 56258893451, 104026172333, 192250283436, 355128933965, 655721712863, 1210278752341, 2233057038717, 4118864374182, 7595069578120, 14001503585962, 25805790761057, 47552037269746, 87607136926107, 161375029879775, 297212439318798, 547315857570701, 1007756129749523, 1855344748636447, 3415469444796671, 6286909205453662, 11571480425293840, 21296546270952008, 39192336845540319, 72121999689790602, 132712424628910003, 244193991874410181, 449303746196226824, 826663333622689889, 1520906740299763239, 2798100437538294325, 5147687743057010881, 9470011493371194242, 17421251147069349520, 32047908821567239498, 58953889434701488165, 108447224268113346054, 199488721877384096731, 366955014767521151335, 674997950938420815410, 1241617273250673411697, 2283858342589065832547, 4200946200472281653279, 7727197025972617621307, 14213272829117031892058, 26143500414367967819088, 48087387194137442381240, 88449760915961398548939, 162689826428129406621518, 299242012615726713002907, 550406236137119308986317, 1012378429327413323486980, 1862092768491777397267141, 3424985658493681981121615, 6299634051328957075863781, 11587002557914483155151877, 21312097082467078033370894, 39199510786200955908396056, 72099882076077130430384922, 132613570629164850339727177, 243916367506066090837872426, 448635388531494401583346875, 825174434303387171602418943, 1517741085220791657395641878, 2791575264797027232056798661
Generating function in Maple syntax:
-(x^14+3*x^13-x^12-12*x^11-13*x^10+6*x^9+24*x^8+11*x^7-11*x^6-9*x^5-2*x^4+4*x^3+4*x^2-4*x+1)/(x^3+x^2+x-1)/(x-1)^2/(x^2+x-1)^2
Generating function in latex syntax:
-\frac{x^{14}+3 x^{13}-x^{12}-12 x^{11}-13 x^{10}+6 x^{9}+24 x^{8}+11 x^{7}-11 x^{6}-9 x^{5}-2 x^{4}+4 x^{3}+4 x^{2}-4 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(-x**14 - 3*x**13 + x**12 + 12*x**11 + 13*x**10 - 6*x**9 - 24*x**8 - 11*x**7 + 11*x**6 + 9*x**5 + 2*x**4 - 4*x**3 - 4*x**2 + 4*x - 1)/((x - 1)**2*(x**2 + x - 1)**2*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x-1)^2*(x^2+x-1)^2*F(x)+x^14+3*x^13-x^12-12*x^11-13*x^10+6*x^9+24*x^8+11*x^7-11*x^6-9*x^5-2*x^4+4*x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+x^{14}+3 x^{13}-x^{12}-12 x^{11}-13 x^{10}+6 x^{9}+24 x^{8}+11 x^{7}-11 x^{6}-9 x^{5}-2 x^{4}+4 x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,n = 4,19,n = 5,50,1/13200*(((28325*I+4575*11^(1/2))*3^(1/2)-13725*I*11^(1/2)-28325)*(17+3*11^(1/2)*3^(1/2))^(2/3)+41800+((14300*I-3600*11^(1/2))*3^(1/2)-10800*I*11^(1/2)+14300)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/13200*(((-28325*I+4575*11^(1/2))*3^(1/2)+13725*I*11^(1/2)-28325)*(17+3*11^(1/2)*3^(1/2))^(2/3)+41800+((-14300*I-3600*11^(1/2))*3^(1/2)+10800*I*11^(1/2)+14300)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/13200*((-9150*11^(1/2)*3^(1/2)+56650)*(17+3*11^(1/2)*3^(1/2))^(2/3)+7200*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-28600*(17+3*11^(1/2)*3^(1/2))^(1/3)+41800)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/13200*((3960*n+16632)*5^(1/2)-9240*n-33000)*(-1/2*5^(1/2)-1/2)^(-n)+1/13200*((-3960*n-16632)*5^(1/2)-9240*n-33000)*(1/2*5^(1/2)-1/2)^(-n)-n-21/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 19 & n =4 \\ 50 & n =5 \\ \frac{\left(\left(\left(28325 \,\mathrm{I}+4575 \sqrt{11}\right) \sqrt{3}-13725 \,\mathrm{I} \sqrt{11}-28325\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+41800+\left(\left(14300 \,\mathrm{I}-3600 \sqrt{11}\right) \sqrt{3}-10800 \,\mathrm{I} \sqrt{11}+14300\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(\left(-28325 \,\mathrm{I}+4575 \sqrt{11}\right) \sqrt{3}+13725 \,\mathrm{I} \sqrt{11}-28325\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+41800+\left(\left(-14300 \,\mathrm{I}-3600 \sqrt{11}\right) \sqrt{3}+10800 \,\mathrm{I} \sqrt{11}+14300\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(-9150 \sqrt{11}\, \sqrt{3}+56650\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+7200 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-28600 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+41800\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{13200}\\+\frac{\left(\left(3960 n +16632\right) \sqrt{5}-9240 n -33000\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}+\\\frac{\left(\left(-3960 n -16632\right) \sqrt{5}-9240 n -33000\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}-n -\frac{21}{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 111
a(7) = 231
a(8) = 470
a(9) = 944
a(10) = 1880
a(11) = 3719
a(12) = 7314
a(13) = 14307
a(14) = 27845
a(n+4) = 1/4*a(n)+3/4*a(n+1)+1/2*a(n+2)-1/2*a(n+3)+3/4*a(n+6)-1/4*a(n+7)+1/2*n+17/4, n >= 15
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) = 50
a \! \left(6\right) = 111
a \! \left(7\right) = 231
a \! \left(8\right) = 470
a \! \left(9\right) = 944
a \! \left(10\right) = 1880
a \! \left(11\right) = 3719
a \! \left(12\right) = 7314
a \! \left(13\right) = 14307
a \! \left(14\right) = 27845
a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+\frac{n}{2}+\frac{17}{4}, \quad n \geq 15
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16592/
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[19,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[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[14,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[20,x]+F[57,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[13,x]+F[28,x]+F[53,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[16,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[28,x]
F[35,x] = 2*F[13,x]+F[36,x]+F[40,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[31,x]
F[39,x] = F[36,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[4,x]+F[46,x]
F[46,x] = F[32,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = 2*F[13,x]+F[40,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[46,x]
F[52,x] = F[49,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[20,x]+F[55,x]
F[55,x] = F[13,x]+F[43,x]+F[56,x]
F[56,x] = F[20,x]*F[4,x]
F[57,x] = F[13,x]+F[58,x]+F[98,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[81,x]
F[60,x] = F[61,x]+F[7,x]
F[61,x] = F[13,x]+F[32,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[24,x]+F[65,x]
F[65,x] = F[13,x]+F[32,x]+F[66,x]
F[66,x] = F[24,x]*F[4,x]
F[67,x] = F[68,x]+F[76,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[73,x]
F[71,x] = F[24,x]+F[72,x]
F[72,x] = F[66,x]
F[73,x] = F[68,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[68,x]
F[76,x] = 2*F[13,x]+F[75,x]+F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[46,x]
F[80,x] = F[77,x]
F[81,x] = F[82,x]+F[93,x]
F[82,x] = F[83,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[87,x]
F[85,x] = F[7,x]+F[86,x]
F[86,x] = F[32,x]
F[87,x] = F[82,x]+F[88,x]
F[88,x] = 2*F[13,x]+F[40,x]+F[89,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[86,x]
F[92,x] = F[89,x]
F[93,x] = 2*F[13,x]+F[40,x]+F[94,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[96,x]+F[97,x]
F[96,x] = F[61,x]
F[97,x] = F[94,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[100,x]+F[111,x]
F[100,x] = F[101,x]+F[20,x]
F[101,x] = F[102,x]+F[13,x]+F[56,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[104,x]+F[105,x]
F[104,x] = F[4,x]+F[65,x]
F[105,x] = F[106,x]+F[42,x]
F[106,x] = 2*F[13,x]+F[107,x]+F[40,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = F[65,x]
F[110,x] = F[107,x]
F[111,x] = F[112,x]+F[120,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[117,x]
F[115,x] = F[116,x]+F[20,x]
F[116,x] = F[56,x]
F[117,x] = F[112,x]+F[118,x]
F[118,x] = F[119,x]
F[119,x] = F[112,x]*F[4,x]
F[120,x] = 2*F[13,x]+F[119,x]+F[121,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[123,x]+F[124,x]
F[123,x] = F[42,x]
F[124,x] = F[121,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_{19}\! \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_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{57}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
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_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)+F_{53}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{31}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \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_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right)+F_{58}\! \left(x \right)+F_{98}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{81}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{7}\! \left(x \right)
F_{61}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \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_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{76}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \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_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{24}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{66}\! \left(x \right)
F_{73}\! \left(x \right) = F_{68}\! \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_{68}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{46}\! \left(x \right)
F_{80}\! \left(x \right) = F_{77}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{93}\! \left(x \right)
F_{82}\! \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_{87}\! \left(x \right)
F_{85}\! \left(x \right) = F_{7}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{32}\! \left(x \right)
F_{87}\! \left(x \right) = F_{82}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{92}\! \left(x \right)
F_{91}\! \left(x \right) = F_{86}\! \left(x \right)
F_{92}\! \left(x \right) = F_{89}\! \left(x \right)
F_{93}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{97}\! \left(x \right)
F_{96}\! \left(x \right) = F_{61}\! \left(x \right)
F_{97}\! \left(x \right) = F_{94}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{111}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{20}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{13}\! \left(x \right)+F_{56}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{105}\! \left(x \right)
F_{104}\! \left(x \right) = F_{4}\! \left(x \right)+F_{65}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{42}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{107}\! \left(x \right)+F_{40}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = F_{65}\! \left(x \right)
F_{110}\! \left(x \right) = F_{107}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{120}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{117}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{20}\! \left(x \right)
F_{116}\! \left(x \right) = F_{56}\! \left(x \right)
F_{117}\! \left(x \right) = F_{112}\! \left(x \right)+F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{112}\! \left(x \right) F_{4}\! \left(x \right)
F_{120}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{119}\! \left(x \right)+F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{124}\! \left(x \right)
F_{123}\! \left(x \right) = F_{42}\! \left(x \right)
F_{124}\! \left(x \right) = F_{121}\! \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_19(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_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_20(x) + F_57(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_13(x) + F_28(x) + F_53(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_16(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_28(x))
Eq(F_35(x), 2*F_13(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_31(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_4(x) + F_46(x))
Eq(F_46(x), F_32(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), 2*F_13(x) + F_40(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_46(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_20(x) + F_55(x))
Eq(F_55(x), F_13(x) + F_43(x) + F_56(x))
Eq(F_56(x), F_20(x)*F_4(x))
Eq(F_57(x), F_13(x) + F_58(x) + F_98(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_81(x))
Eq(F_60(x), F_61(x) + F_7(x))
Eq(F_61(x), F_13(x) + F_32(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_24(x) + F_65(x))
Eq(F_65(x), F_13(x) + F_32(x) + F_66(x))
Eq(F_66(x), F_24(x)*F_4(x))
Eq(F_67(x), F_68(x) + F_76(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_73(x))
Eq(F_71(x), F_24(x) + F_72(x))
Eq(F_72(x), F_66(x))
Eq(F_73(x), F_68(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_68(x))
Eq(F_76(x), 2*F_13(x) + F_75(x) + F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_46(x))
Eq(F_80(x), F_77(x))
Eq(F_81(x), F_82(x) + F_93(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_87(x))
Eq(F_85(x), F_7(x) + F_86(x))
Eq(F_86(x), F_32(x))
Eq(F_87(x), F_82(x) + F_88(x))
Eq(F_88(x), 2*F_13(x) + F_40(x) + F_89(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_92(x))
Eq(F_91(x), F_86(x))
Eq(F_92(x), F_89(x))
Eq(F_93(x), 2*F_13(x) + F_40(x) + F_94(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_97(x))
Eq(F_96(x), F_61(x))
Eq(F_97(x), F_94(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_111(x))
Eq(F_100(x), F_101(x) + F_20(x))
Eq(F_101(x), F_102(x) + F_13(x) + F_56(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_105(x))
Eq(F_104(x), F_4(x) + F_65(x))
Eq(F_105(x), F_106(x) + F_42(x))
Eq(F_106(x), F_107(x) + 2*F_13(x) + F_40(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_65(x))
Eq(F_110(x), F_107(x))
Eq(F_111(x), F_112(x) + F_120(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_117(x))
Eq(F_115(x), F_116(x) + F_20(x))
Eq(F_116(x), F_56(x))
Eq(F_117(x), F_112(x) + F_118(x))
Eq(F_118(x), F_119(x))
Eq(F_119(x), F_112(x)*F_4(x))
Eq(F_120(x), F_119(x) + F_121(x) + 2*F_13(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_123(x) + F_124(x))
Eq(F_123(x), F_42(x))
Eq(F_124(x), F_121(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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [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, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": 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