0132_0231_0321_1302_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 159, 445, 1238, 3435, 9520, 26372, 73041, 202282, 560189, 1551339, 4296124, 11897237, 32946942, 91239730, 252669503, 699715740, 1937717467, 5366106185, 14860316850, 41152561855, 113963474924, 315598179792, 873983626333, 2420316174158, 6702563075961, 18561356680087, 51401823138488, 142346675811369, 394199560975738, 1091653830254366, 3023108605598763, 8371871547517696, 23184159867208727, 64203716659707285, 177798861659430862, 492377028185794067, 1363535939557658664, 3776029652146612828, 10456930045067973929, 28958296422614967666, 80193988874923541941, 222080600247223632131, 615006108289362322164, 1703131713495786876349, 4716469632445221939062, 13061283292129241263050, 36170511957438288726487, 100166722212630846855524, 277390937972583493103187, 768176603662557921180577, 2127305595227721117294122, 5891131119993188923805031, 16314264368414403575301604, 45178967580477742919393960, 125113769493012014106685365, 346476605266972702975963798, 959495013888383837740600049, 2657122205891190824352521775, 7358348209052161338292929136, 20377417435153677028266865745, 56431026302300060570860189938, 156274009680798666022606633270, 432768420175249614525226575683, 1198462277147250238123933283784, 3318892420947299765869179909007, 9190983405869898674008833018989, 25452520073809646494281252431814, 70485469236506340574704104185531, 195194871041579273903839893143840, 540551663962041127631096827677876, 1496945590081050449564154694075393, 4145479977322689494965207147692218, 11480046005849059162962563576679213, 31791603630305538443486638276420891, 88040244862390245662462558400275756, 243809176962714257706863600501085669, 675178889656058065901614881379402222, 1869767736867850185660443602037396834, 5177933497910206748786579525233874031, 14339211646518828126600909855043627564, 39709469178514127816676593641934405579, 109967129386933762072215450595993463289, 304531130628805986526570668001089611938, 843335731677998325324173147049209778191, 2335443193792122751518164223742533186012, 6467524980327175915756890192179479391872, 17910467478867403321076679499845114767885, 49599320650067156798991312531098398097886, 137355019451661242991654148285629558917737, 380376205183783975497047811825635393423303, 1053372916749757840298480599722375019450152, 2917097564517150788390048135627119223844985, 8078295981985478500368711618984618045508202, 22371163298189042553014567321049109932129870, 61952291477098799947065038479789830974726491
Generating function in Maple syntax:
(x^5-2*x^4+2*x^3-5*x^2+4*x-1)/(x^3-x^2+3*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{5}-2 x^{4}+2 x^{3}-5 x^{2}+4 x -1}{\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 2*x**4 + 2*x**3 - 5*x**2 + 4*x - 1)/((x - 1)**2*(x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-x^2+3*x-1)*(x-1)^2*F(x)-x^5+2*x^4-2*x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{5}+2 x^{4}-2 x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,3/256*(((I+11/57*19^(1/2))*3^(1/2)-11/19*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(1/3)+(-4*I-20/57*19^(1/2))*3^(1/2)-20/19*I*19^(1/2)-4)*(1+3*19^(1/2)*3^(1/2))^(1/3)*(1/384*((I+3*19^(1/2))*3^(1/2)-9*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-1/6*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3)^(-n)-3/256*(((I-11/57*19^(1/2))*3^(1/2)-11/19*I*19^(1/2)+1)*(1+3*19^(1/2)*3^(1/2))^(1/3)+(-4*I+20/57*19^(1/2))*3^(1/2)-20/19*I*19^(1/2)+4)*(1+3*19^(1/2)*3^(1/2))^(1/3)*(1/384*((-I+3*19^(1/2))*3^(1/2)+9*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-1/6*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3)^(-n)+1/4864*(40*(1+3*19^(1/2)*3^(1/2))^(1/3)*19^(1/2)*3^(1/2)-22*(1+3*19^(1/2)*3^(1/2))^(2/3)*19^(1/2)*3^(1/2)+456*(1+3*19^(1/2)*3^(1/2))^(1/3)+114*(1+3*19^(1/2)*3^(1/2))^(2/3))*(1/3*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3+1/192*(1+3*19^(1/2)*3^(1/2))^(2/3)-1/64*(1+3*19^(1/2)*3^(1/2))^(2/3)*19^(1/2)*3^(1/2))^(-n)-1/2*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{3 \left(\left(\left(\mathrm{I}+\frac{11 \sqrt{19}}{57}\right) \sqrt{3}-\frac{11 \,\mathrm{I} \sqrt{19}}{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-4 \,\mathrm{I}-\frac{20 \sqrt{19}}{57}\right) \sqrt{3}-\frac{20 \,\mathrm{I} \sqrt{19}}{19}-4\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\frac{\left(\left(\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{256}\\-\\\frac{3 \left(\left(\left(\mathrm{I}-\frac{11 \sqrt{19}}{57}\right) \sqrt{3}-\frac{11 \,\mathrm{I} \sqrt{19}}{19}+1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-4 \,\mathrm{I}+\frac{20 \sqrt{19}}{57}\right) \sqrt{3}-\frac{20 \,\mathrm{I} \sqrt{19}}{19}+4\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\frac{\left(\left(-\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{256}\\+\\\frac{\left(40 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{19}\, \sqrt{3}-22 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}+456 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+114 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}}{64}\right)^{-n}}{4864}\\-\frac{n}{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) = 56
a(n+3) = a(n)-a(n+1)+3*a(n+2)+n+1, n >= 6
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 56
a \! \left(n +3\right) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+n +1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18688/
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[12,x]*F[4,x]
F[4,x] = F[18,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[41,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[9,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[19,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[19,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[33,x]
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[19,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[36,x]
F[41,x] = F[15,x]+F[42,x]+F[76,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[55,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[16,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[51,x]
F[55,x] = F[41,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[71,x]
F[59,x] = F[60,x]+F[69,x]
F[60,x] = F[15,x]+F[61,x]+F[67,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[10,x]+F[64,x]
F[64,x] = F[46,x]
F[65,x] = F[60,x]+F[66,x]
F[66,x] = F[57,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[19,x]+F[60,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[59,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[60,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[71,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[68,x]+F[78,x]
F[78,x] = F[79,x]+F[96,x]
F[79,x] = F[15,x]+F[38,x]+F[80,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[82,x]+F[89,x]
F[82,x] = F[12,x]+F[83,x]
F[83,x] = F[84,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[86,x]
F[86,x] = F[12,x]+F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[12,x]*F[86,x]
F[89,x] = F[37,x]+F[90,x]
F[90,x] = F[91,x]
F[91,x] = F[12,x]*F[92,x]
F[92,x] = F[93,x]
F[93,x] = F[37,x]+F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[12,x]*F[93,x]
F[96,x] = F[113,x]+F[15,x]+F[73,x]+F[97,x]
F[97,x] = F[12,x]*F[98,x]
F[98,x] = F[106,x]+F[99,x]
F[99,x] = F[100,x]+F[52,x]
F[100,x] = F[101,x]
F[101,x] = F[102,x]*F[12,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]+F[52,x]
F[104,x] = F[105,x]
F[105,x] = F[103,x]*F[12,x]
F[106,x] = F[107,x]+F[72,x]
F[107,x] = F[108,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = F[110,x]
F[110,x] = F[111,x]+F[72,x]
F[111,x] = F[112,x]
F[112,x] = F[110,x]*F[12,x]
F[113,x] = F[12,x]*F[78,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{76}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{55}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{71}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{69}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{46}\! \left(x \right)
F_{65}\! \left(x \right) = F_{60}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{57}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{19}\! \left(x \right)+F_{60}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{68}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{96}\! \left(x \right)
F_{79}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{89}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{12}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{89}\! \left(x \right) = F_{37}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{37}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{96}\! \left(x \right) = F_{113}\! \left(x \right)+F_{15}\! \left(x \right)+F_{73}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{106}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{52}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{12}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{52}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{103}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{72}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{72}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_18(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_41(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_9(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_19(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_19(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_33(x))
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_19(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_36(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_76(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_55(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_16(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_51(x))
Eq(F_55(x), F_41(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_71(x))
Eq(F_59(x), F_60(x) + F_69(x))
Eq(F_60(x), F_15(x) + F_61(x) + F_67(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_10(x) + F_64(x))
Eq(F_64(x), F_46(x))
Eq(F_65(x), F_60(x) + F_66(x))
Eq(F_66(x), F_57(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_19(x) + F_60(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_59(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_60(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_71(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_68(x) + F_78(x))
Eq(F_78(x), F_79(x) + F_96(x))
Eq(F_79(x), F_15(x) + F_38(x) + F_80(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_89(x))
Eq(F_82(x), F_12(x) + F_83(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_12(x) + F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_12(x)*F_86(x))
Eq(F_89(x), F_37(x) + F_90(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_12(x)*F_92(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_37(x) + F_94(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_12(x)*F_93(x))
Eq(F_96(x), F_113(x) + F_15(x) + F_73(x) + F_97(x))
Eq(F_97(x), F_12(x)*F_98(x))
Eq(F_98(x), F_106(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_52(x))
Eq(F_100(x), F_101(x))
Eq(F_101(x), F_102(x)*F_12(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x) + F_52(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_103(x)*F_12(x))
Eq(F_106(x), F_107(x) + F_72(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_111(x) + F_72(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_110(x)*F_12(x))
Eq(F_113(x), F_12(x)*F_78(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 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]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 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]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 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]]}], "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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 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