0132_0231_0312_1032_2130
Counting sequence:
1, 1, 2, 6, 19, 57, 166, 481, 1393, 4032, 11667, 33757, 97670, 282589, 817613, 2365592, 6844343, 19802665, 57294838, 165770537, 479622105, 1387685456, 4014975339, 11616484773, 33609849894, 97243015589, 281352166421, 814033183464, 2355233415151, 6814371394833, 19715947136278, 57043936844305, 165044606186305, 477521775987168, 1381608595463811, 3997393222776877, 11565614624841158, 33462667842674221, 96817175348735261, 280120087453213880, 810468422696254823, 2344919531333021113, 6784530346209236278, 19629608352687197369, 56794133774518398057, 164321853663286469360, 475430643920715404763, 1375558345647896541045, 3979888100349547157094, 11514967243242175739765, 33316130320672773144293, 96393199919475238007304, 278893403924237292357151, 806919272494582185352737, 2334650813398459739715670, 6754819975549271618508577, 19543647658238519486515537, 56545424625963206320469376, 163602266170749642711909747, 473348669202762760788290941, 1369534590689465052293901830, 3962459635207131681517234045, 11464541653337455513834382189, 33170234506184164964573716184, 95971081135629302742921234455, 277672092207378654291858751561, 803385664498874990061802452598, 2324427063560504921379391075145, 6725239710597397082660088275961, 19458063397229436306724478657936, 56297804608211921680272129848139, 162885829848601122899412087438405, 471275811732050319631413089782950, 1363537214563973679355371441606725, 3945107491657072964233525409914037, 11414336883871345852608030967701288, 33024977589084911952301310376403279, 95550810866697535932762950981336625, 276456128778750387551203604503009174, 799867530647723450625450483280497457, 2314248084897815036190673747301156449, 6695788981603222433048977967628990240, 19372853921207299164207161122146809187, 56051268951814305131953852893413750221, 162172530897442061097752353493352421958, 469212031582926476489717529830937134029, 1357566101754965978635307941786286480573, 3927831335476855581611711002514627151608, 11364351967841453719179260125419469242311, 32880356771557437533196685257316353536473, 95132381017784570043634217651558845670326, 275245490217479180036064487948199121959129, 796364803177767845130952616707671227280009, 2304113681351393495444061797477932251221552, 6666467221311371001273361751622523770302907, 19288017588938255198637928690804981514247189, 55805812908206181585528547915748285274881766, 161462355578303031560494438559684921851003925, 467157289004579423493230625144916443306624389, 1351621137251847602090254532706560978618632776, 3910630833907569445898521850094136336251281791
Generating function in Maple syntax:
(x^5+2*x^4-x^3+2*x^2-3*x+1)/(x-1)/(2*x^3-x^2+3*x-1)
Generating function in latex syntax:
\frac{x^{5}+2 x^{4}-x^{3}+2 x^{2}-3 x +1}{\left(x -1\right) \left(2 x^{3}-x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(x**5 + 2*x**4 - x**3 + 2*x**2 - 3*x + 1)/((x - 1)*(2*x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x^3-x^2+3*x-1)*F(x)+x^5+2*x^4-x^3+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}-x^{3}+2 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/258264*(((-175*211^(1/2)-3376*I)*3^(1/2)-525*I*211^(1/2)-3376)*(28+3*211^(1/2)*3^(1/2))^(1/3)-7174+((52*211^(1/2)+1477*I)*3^(1/2)-156*I*211^(1/2)-1477)*(28+3*211^(1/2)*3^(1/2))^(2/3))*(1/3468*((28*I+3*211^(1/2))*3^(1/2)-9*I*211^(1/2)-28)*(28+3*211^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(28+3*211^(1/2)*3^(1/2))^(1/3)-1/12*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6)^(-n)+1/258264*(((52*211^(1/2)-1477*I)*3^(1/2)+156*I*211^(1/2)-1477)*(28+3*211^(1/2)*3^(1/2))^(2/3)-7174+((-175*211^(1/2)+3376*I)*3^(1/2)+525*I*211^(1/2)-3376)*(28+3*211^(1/2)*3^(1/2))^(1/3))*(1/3468*((-28*I+3*211^(1/2))*3^(1/2)+9*I*211^(1/2)-28)*(28+3*211^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(28+3*211^(1/2)*3^(1/2))^(1/3)-1/12*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6)^(-n)-2/3+1/258264*((-104*211^(1/2)*3^(1/2)+2954)*(28+3*211^(1/2)*3^(1/2))^(2/3)+350*(28+3*211^(1/2)*3^(1/2))^(1/3)*211^(1/2)*3^(1/2)+6752*(28+3*211^(1/2)*3^(1/2))^(1/3)-7174)*(1/6*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6+14/867*(28+3*211^(1/2)*3^(1/2))^(2/3)-1/578*(28+3*211^(1/2)*3^(1/2))^(2/3)*211^(1/2)*3^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-175 \sqrt{211}-3376 \,\mathrm{I}\right) \sqrt{3}-525 \,\mathrm{I} \sqrt{211}-3376\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}-7174+\left(\left(52 \sqrt{211}+1477 \,\mathrm{I}\right) \sqrt{3}-156 \,\mathrm{I} \sqrt{211}-1477\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}-\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{258264}\\+\\\frac{\left(\left(\left(52 \sqrt{211}-1477 \,\mathrm{I}\right) \sqrt{3}+156 \,\mathrm{I} \sqrt{211}-1477\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}-7174+\left(\left(-175 \sqrt{211}+3376 \,\mathrm{I}\right) \sqrt{3}+525 \,\mathrm{I} \sqrt{211}-3376\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}+\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{258264}\\-\frac{2}{3}+\\\frac{\left(\left(-104 \sqrt{211}\, \sqrt{3}+2954\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}+350 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{211}\, \sqrt{3}+6752 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}-7174\right) \left(\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}+\frac{14 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{867}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}}{578}\right)^{-n}}{258264} & \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) = 57
a(n+3) = 2*a(n)-a(n+1)+3*a(n+2)+2, 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) = 57
a \! \left(n +3\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18593/
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[20,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[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[17,x]
F[20,x] = F[21,x]+F[40,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[29,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[14,x]
F[29,x] = F[21,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[21,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[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[21,x]+F[37,x]
F[40,x] = F[41,x]+F[42,x]+F[69,x]
F[41,x] = 0
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[53,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[41,x]+F[46,x]+F[52,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[17,x]
F[52,x] = F[12,x]*F[13,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[41,x]+F[42,x]+F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[21,x]+F[54,x]
F[57,x] = 2*F[41,x]+F[58,x]+F[67,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[54,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[60,x]
F[63,x] = F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[54,x]+F[64,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[37,x]+F[64,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[56,x]+F[71,x]
F[71,x] = F[72,x]+F[95,x]
F[72,x] = F[41,x]+F[73,x]+F[93,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[75,x]+F[83,x]
F[75,x] = F[12,x]+F[76,x]
F[76,x] = F[41,x]+F[77,x]+F[82,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]+F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[12,x]*F[79,x]
F[82,x] = F[12,x]*F[14,x]
F[83,x] = F[84,x]+F[86,x]
F[84,x] = F[85,x]
F[85,x] = F[12,x]*F[21,x]
F[86,x] = 2*F[41,x]+F[87,x]+F[92,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[84,x]+F[90,x]
F[90,x] = F[91,x]
F[91,x] = F[12,x]*F[89,x]
F[92,x] = F[12,x]*F[37,x]
F[93,x] = F[12,x]*F[94,x]
F[94,x] = F[21,x]+F[72,x]
F[95,x] = 2*F[41,x]+F[118,x]+F[96,x]
F[96,x] = F[12,x]*F[97,x]
F[97,x] = F[108,x]+F[98,x]
F[98,x] = F[101,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[10,x]*F[12,x]
F[101,x] = 2*F[41,x]+F[102,x]+F[107,x]
F[102,x] = F[103,x]*F[12,x]
F[103,x] = F[104,x]
F[104,x] = F[105,x]+F[99,x]
F[105,x] = F[106,x]
F[106,x] = F[104,x]*F[12,x]
F[107,x] = F[12,x]*F[17,x]
F[108,x] = F[109,x]+F[111,x]
F[109,x] = F[110,x]
F[110,x] = F[12,x]*F[54,x]
F[111,x] = 3*F[41,x]+F[112,x]+F[117,x]
F[112,x] = F[113,x]*F[12,x]
F[113,x] = F[114,x]
F[114,x] = F[109,x]+F[115,x]
F[115,x] = F[116,x]
F[116,x] = F[114,x]*F[12,x]
F[117,x] = F[12,x]*F[64,x]
F[118,x] = F[119,x]*F[12,x]
F[119,x] = F[54,x]+F[95,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_{20}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \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_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \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_{21}\! \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_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{69}\! \left(x \right)
F_{41}\! \left(x \right) = 0
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_{53}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \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_{17}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{21}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{58}\! \left(x \right)+F_{67}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{54}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \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_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{54}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{37}\! \left(x \right)+F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{56}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{95}\! \left(x \right)
F_{72}\! \left(x \right) = F_{41}\! \left(x \right)+F_{73}\! \left(x \right)+F_{93}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{83}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{41}\! \left(x \right)+F_{77}\! \left(x \right)+F_{82}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{87}\! \left(x \right)+F_{92}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{84}\! \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_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{21}\! \left(x \right)+F_{72}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{118}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{12}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{108}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{101}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{101}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{102}\! \left(x \right)+F_{107}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{12}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{99}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{107}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{111}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{111}\! \left(x \right) = 3 F_{41}\! \left(x \right)+F_{112}\! \left(x \right)+F_{117}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)
F_{114}\! \left(x \right) = F_{109}\! \left(x \right)+F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)
F_{116}\! \left(x \right) = F_{114}\! \left(x \right) F_{12}\! \left(x \right)
F_{117}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{54}\! \left(x \right)+F_{95}\! \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_20(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_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_40(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_29(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_14(x))
Eq(F_29(x), F_21(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_21(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))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_21(x) + F_37(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_69(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_53(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_41(x) + F_46(x) + F_52(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_17(x))
Eq(F_52(x), F_12(x)*F_13(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_41(x) + F_42(x) + F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_21(x) + F_54(x))
Eq(F_57(x), 2*F_41(x) + F_58(x) + F_67(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_54(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_60(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_54(x) + F_64(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_37(x) + F_64(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_56(x) + F_71(x))
Eq(F_71(x), F_72(x) + F_95(x))
Eq(F_72(x), F_41(x) + F_73(x) + F_93(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_83(x))
Eq(F_75(x), F_12(x) + F_76(x))
Eq(F_76(x), F_41(x) + F_77(x) + F_82(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x) + F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_12(x)*F_79(x))
Eq(F_82(x), F_12(x)*F_14(x))
Eq(F_83(x), F_84(x) + F_86(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_12(x)*F_21(x))
Eq(F_86(x), 2*F_41(x) + F_87(x) + F_92(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_84(x) + F_90(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_12(x)*F_89(x))
Eq(F_92(x), F_12(x)*F_37(x))
Eq(F_93(x), F_12(x)*F_94(x))
Eq(F_94(x), F_21(x) + F_72(x))
Eq(F_95(x), F_118(x) + 2*F_41(x) + F_96(x))
Eq(F_96(x), F_12(x)*F_97(x))
Eq(F_97(x), F_108(x) + F_98(x))
Eq(F_98(x), F_101(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_10(x)*F_12(x))
Eq(F_101(x), F_102(x) + F_107(x) + 2*F_41(x))
Eq(F_102(x), F_103(x)*F_12(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_105(x) + F_99(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_104(x)*F_12(x))
Eq(F_107(x), F_12(x)*F_17(x))
Eq(F_108(x), F_109(x) + F_111(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_12(x)*F_54(x))
Eq(F_111(x), F_112(x) + F_117(x) + 3*F_41(x))
Eq(F_112(x), F_113(x)*F_12(x))
Eq(F_113(x), F_114(x))
Eq(F_114(x), F_109(x) + F_115(x))
Eq(F_115(x), F_116(x))
Eq(F_116(x), F_114(x)*F_12(x))
Eq(F_117(x), F_12(x)*F_64(x))
Eq(F_118(x), F_119(x)*F_12(x))
Eq(F_119(x), F_54(x) + F_95(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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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], [1, 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, 0, 1], "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], [0, 0], [1, 0]]}, {"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, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 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], [2, 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, 0, 1], "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], [0, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 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, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [1, 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, 0, 1], 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