0132_0231_0321_3012
Counting sequence:
1, 1, 2, 6, 20, 65, 207, 654, 2061, 6490, 20432, 64320, 202475, 637373, 2006388, 6315909, 19881846, 62586046, 197014556, 620181933, 1952270111, 6145549202, 19345568409, 60897896142, 191700428544, 603453594160, 1899610987151, 5979783594669, 18823755011384, 59255280248681, 186529639555242, 587176472486310, 1848372251528196, 5818489228209097, 18316016630738111, 57656971089853350, 181498323695408165, 571338398142810722, 1798515592574871744, 5661545499566462976, 17821973618684047395, 56101773568611672397, 176602718917961010380, 555927528584693071725, 1750003731153500928158, 5508835057777885507838, 17341256566235732318604, 54588524822762773679061, 171839164650121783268271, 540932340698434544354506, 1702800393666135601292273, 5360243717219360555373846, 16873506028577469961973072, 53116093356317933783063024, 167204099056758685801706727, 526341622186658411819610909, 1656870284932675523656386480, 5215660372227485544968419921, 16418372256298574020724131154, 51683378193444514971647651334, 162694056377109823369386631796, 512144463184404322583327704337, 1612179061799732335196798573295, 5074976914022826196406597811966, 15975514933876512102619323232317, 50289308055237302156454298066666, 158305664333342498514365820381680, 498330248101534599690719710559040, 1568693307461348205376235679157787, 4938088149874131520947149405335709, 15544602925214016067350610294428516, 48932840558695017911937228640864725, 154035641609012154966644233085509062, 484888647685177582198115856713870590, 1526380506472368881804467956239819900, 4804891724442954935590168487903176893, 15125314026039890679145385939607118463, 47612961437303066791615506250958763074, 149880795395535550064616035672916873801, 471809611296272890562763009878398324446, 1485209020435787978848819438745801439456, 4675288043249868793771604761020927658288, 14717334722988384129102723737724704639231, 46328683782639718113944900106581837021517, 145838019004843588862243033974174161826664, 459083359394443585236240159314675840881337, 1445148064345883292472176435442762715066874, 4549180198205046003042228297801197499450150, 14320359959176977719988624044191177693481316, 45079047306438656163299987718803880990641689, 141904289546428737496276307040443188381572959, 446700376225575959215193889049552288247848086, 1406167683569456280535246622795683468837315157, 4426473895147525981991681131074520198987289330, 13934092906107311045576178544960582677685059936, 43863117622556122516864917409414453582615686912, 138076665667050082912722815133850586496394857171, 434651402706639275145872686852119567915417427885, 1368238731447962974362187219416340965755100826940, 4307077383338984154923853632242855745770545881277, 13558244741718686971754725557574519653732298472174
Generating function in Maple syntax:
(x^4-x^3+4*x^2-4*x+1)/(x-1)/(x^3-3*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{4}-x^{3}+4 x^{2}-4 x +1}{\left(x -1\right) \left(x^{3}-3 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**4 - x**3 + 4*x**2 - 4*x + 1)/((x - 1)*(x**3 - 3*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-3*x^2+4*x-1)*F(x)-x^4+x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-x^{4}+x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/2232*(39*((I*3^(1/3)-1/3*3^(5/6))*31^(1/2)-93/13*I*3^(5/6)+93/13*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)+744-81*2^(1/3)*((I*3^(2/3)+3^(1/6))*31^(1/2)-217/27*I*3^(1/6)-217/81*3^(2/3))*(9+31^(1/2)*3^(1/2))^(2/3))*(1/12*(108+12*31^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*31^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1+1/24*((-I*31^(1/2)+3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)-31^(1/2)*3^(1/6)*2^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)+1/2232*(-39*((I*3^(1/3)+1/3*3^(5/6))*31^(1/2)-93/13*I*3^(5/6)-93/13*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)+744+81*2^(1/3)*((I*3^(2/3)-3^(1/6))*31^(1/2)-217/27*I*3^(1/6)+217/81*3^(2/3))*(9+31^(1/2)*3^(1/2))^(2/3))*(1/12*(108+12*31^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*31^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1+1/24*((I*31^(1/2)+3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)-31^(1/2)*3^(1/6)*2^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)-1+1/2232*(26*(31^(1/2)*3^(5/6)-279/13*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)+744+162*2^(1/3)*(3^(1/6)*31^(1/2)-217/81*3^(2/3))*(9+31^(1/2)*3^(1/2))^(2/3))*(-1/6*(108+12*31^(1/2)*3^(1/2))^(1/3)+1+1/12*(31^(1/2)*3^(1/6)*2^(1/3)-3*18^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(39 \left(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{93 \,\mathrm{I} \,3^{\frac{5}{6}}}{13}+\frac{93 \,3^{\frac{1}{3}}}{13}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+744-81 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{31}-\frac{217 \,\mathrm{I} \,3^{\frac{1}{6}}}{27}-\frac{217 \,3^{\frac{2}{3}}}{81}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(-\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2232}\\+\\\frac{\left(-39 \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{93 \,\mathrm{I} \,3^{\frac{5}{6}}}{13}-\frac{93 \,3^{\frac{1}{3}}}{13}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+744+81 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{31}-\frac{217 \,\mathrm{I} \,3^{\frac{1}{6}}}{27}+\frac{217 \,3^{\frac{2}{3}}}{81}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2232}\\-1+\\\frac{\left(26 \left(\sqrt{31}\, 3^{\frac{5}{6}}-\frac{279 \,3^{\frac{1}{3}}}{13}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+744+162 \,2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \sqrt{31}-\frac{217 \,3^{\frac{2}{3}}}{81}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+1+\frac{\left(\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}\right)^{-n}}{2232} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+3) = a(n)-3*a(n+1)+4*a(n+2)+1, n >= 5
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(n +3\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15317/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[11,x]*F[4,x]
F[22,x] = F[23,x]+F[62,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[16,x]*F[4,x]
F[29,x] = F[23,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[60,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[38,x]
F[34,x] = F[27,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[4,x]
F[38,x] = F[30,x]+F[39,x]
F[39,x] = 2*F[31,x]+F[40,x]+F[44,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[35,x]
F[43,x] = F[39,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[31,x]+F[47,x]+F[59,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[4,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = x^2
F[52,x] = F[46,x]+F[53,x]
F[53,x] = 2*F[31,x]+F[54,x]+F[58,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[50,x]
F[57,x] = F[53,x]
F[58,x] = F[4,x]*F[46,x]
F[59,x] = F[23,x]*F[4,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[23,x]+F[46,x]
F[62,x] = F[139,x]+F[31,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[76,x]
F[65,x] = F[66,x]+F[7,x]
F[66,x] = F[31,x]+F[67,x]+F[68,x]
F[67,x] = F[19,x]*F[4,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[20,x]+F[4,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[73,x]
F[73,x] = x^2
F[74,x] = F[75,x]
F[75,x] = F[20,x]*F[4,x]
F[76,x] = F[62,x]+F[77,x]
F[77,x] = F[125,x]+F[132,x]+F[31,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[84,x]
F[80,x] = F[66,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[20,x]
F[84,x] = F[77,x]+F[85,x]
F[85,x] = 2*F[31,x]+F[131,x]+F[86,x]+F[90,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[89,x]
F[88,x] = F[81,x]
F[89,x] = F[85,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[92,x]
F[92,x] = 2*F[31,x]+F[105,x]+F[93,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]+F[98,x]
F[95,x] = F[20,x]+F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[20,x]*F[4,x]
F[98,x] = F[92,x]+F[99,x]
F[99,x] = 3*F[31,x]+F[100,x]+F[104,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[103,x]
F[102,x] = F[96,x]
F[103,x] = F[99,x]
F[104,x] = F[4,x]*F[92,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]+F[129,x]+F[31,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[112,x]
F[109,x] = F[11,x]+F[110,x]
F[110,x] = F[111,x]+F[31,x]+F[67,x]
F[111,x] = F[4,x]*F[70,x]
F[112,x] = F[106,x]+F[113,x]
F[113,x] = F[114,x]+F[125,x]+F[127,x]+F[31,x]
F[114,x] = F[115,x]*F[4,x]
F[115,x] = F[116,x]+F[118,x]
F[116,x] = F[110,x]+F[117,x]
F[117,x] = F[82,x]
F[118,x] = F[113,x]+F[119,x]
F[119,x] = 2*F[31,x]+F[120,x]+F[124,x]+F[90,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[122,x]+F[123,x]
F[122,x] = F[117,x]
F[123,x] = F[119,x]
F[124,x] = 0
F[125,x] = F[126,x]*F[4,x]
F[126,x] = F[106,x]+F[92,x]
F[127,x] = F[128,x]*F[4,x]
F[128,x] = F[46,x]+F[92,x]
F[129,x] = F[130,x]*F[4,x]
F[130,x] = F[106,x]+F[23,x]
F[131,x] = 0
F[132,x] = F[133,x]*F[4,x]
F[133,x] = F[128,x]+F[134,x]
F[134,x] = F[135,x]+F[137,x]
F[135,x] = F[136,x]
F[136,x] = F[4,x]*F[46,x]
F[137,x] = F[138,x]
F[138,x] = F[4,x]*F[92,x]
F[139,x] = F[140,x]*F[4,x]
F[140,x] = F[130,x]+F[141,x]
F[141,x] = F[142,x]+F[144,x]
F[142,x] = F[143,x]+F[31,x]+F[47,x]
F[143,x] = F[4,x]*F[61,x]
F[144,x] = 2*F[31,x]+F[145,x]+F[93,x]
F[145,x] = F[126,x]*F[4,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_{22}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \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_{11}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{62}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{60}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \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_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{59}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \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_{51}\! \left(x \right) = x^{2}
F_{52}\! \left(x \right) = F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{62}\! \left(x \right) = F_{139}\! \left(x \right)+F_{31}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{76}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{7}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{20}\! \left(x \right)+F_{4}\! \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) = x^{2}
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{76}\! \left(x \right) = F_{62}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{125}\! \left(x \right)+F_{132}\! \left(x \right)+F_{31}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{84}\! \left(x \right)
F_{80}\! \left(x \right) = F_{66}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{20}\! \left(x \right)
F_{84}\! \left(x \right) = F_{77}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{131}\! \left(x \right)+F_{86}\! \left(x \right)+F_{90}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{81}\! \left(x \right)
F_{89}\! \left(x \right) = F_{85}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{105}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{98}\! \left(x \right)
F_{95}\! \left(x \right) = F_{20}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{98}\! \left(x \right) = F_{92}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = 3 F_{31}\! \left(x \right)+F_{100}\! \left(x \right)+F_{104}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{103}\! \left(x \right)
F_{102}\! \left(x \right) = F_{96}\! \left(x \right)
F_{103}\! \left(x \right) = F_{99}\! \left(x \right)
F_{104}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{129}\! \left(x \right)+F_{31}\! \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_{112}\! \left(x \right)
F_{109}\! \left(x \right) = F_{11}\! \left(x \right)+F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{111}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{112}\! \left(x \right) = F_{106}\! \left(x \right)+F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{125}\! \left(x \right)+F_{127}\! \left(x \right)+F_{31}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{4}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{118}\! \left(x \right)
F_{116}\! \left(x \right) = F_{110}\! \left(x \right)+F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{82}\! \left(x \right)
F_{118}\! \left(x \right) = F_{113}\! \left(x \right)+F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{120}\! \left(x \right)+F_{124}\! \left(x \right)+F_{90}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{123}\! \left(x \right)
F_{122}\! \left(x \right) = F_{117}\! \left(x \right)
F_{123}\! \left(x \right) = F_{119}\! \left(x \right)
F_{124}\! \left(x \right) = 0
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{106}\! \left(x \right)+F_{92}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right) F_{4}\! \left(x \right)
F_{128}\! \left(x \right) = F_{46}\! \left(x \right)+F_{92}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{4}\! \left(x \right)
F_{130}\! \left(x \right) = F_{106}\! \left(x \right)+F_{23}\! \left(x \right)
F_{131}\! \left(x \right) = 0
F_{132}\! \left(x \right) = F_{133}\! \left(x \right) F_{4}\! \left(x \right)
F_{133}\! \left(x \right) = F_{128}\! \left(x \right)+F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{137}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)
F_{138}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right) F_{4}\! \left(x \right)
F_{140}\! \left(x \right) = F_{130}\! \left(x \right)+F_{141}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{144}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)
F_{143}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{144}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{145}\! \left(x \right)+F_{93}\! \left(x \right)
F_{145}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(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_11(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_11(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_62(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_16(x)*F_4(x))
Eq(F_29(x), F_23(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_60(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_38(x))
Eq(F_34(x), F_27(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_4(x))
Eq(F_38(x), F_30(x) + F_39(x))
Eq(F_39(x), 2*F_31(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_35(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_31(x) + F_47(x) + F_59(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_4(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), x**2)
Eq(F_52(x), F_46(x) + F_53(x))
Eq(F_53(x), 2*F_31(x) + F_54(x) + F_58(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_50(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_4(x)*F_46(x))
Eq(F_59(x), F_23(x)*F_4(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_23(x) + F_46(x))
Eq(F_62(x), F_139(x) + F_31(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_76(x))
Eq(F_65(x), F_66(x) + F_7(x))
Eq(F_66(x), F_31(x) + F_67(x) + F_68(x))
Eq(F_67(x), F_19(x)*F_4(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_20(x) + F_4(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), x**2)
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_20(x)*F_4(x))
Eq(F_76(x), F_62(x) + F_77(x))
Eq(F_77(x), F_125(x) + F_132(x) + F_31(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_84(x))
Eq(F_80(x), F_66(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_20(x))
Eq(F_84(x), F_77(x) + F_85(x))
Eq(F_85(x), F_131(x) + 2*F_31(x) + F_86(x) + F_90(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_89(x))
Eq(F_88(x), F_81(x))
Eq(F_89(x), F_85(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_105(x) + 2*F_31(x) + F_93(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x) + F_98(x))
Eq(F_95(x), F_20(x) + F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_20(x)*F_4(x))
Eq(F_98(x), F_92(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_104(x) + 3*F_31(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_103(x))
Eq(F_102(x), F_96(x))
Eq(F_103(x), F_99(x))
Eq(F_104(x), F_4(x)*F_92(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x) + F_129(x) + F_31(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_112(x))
Eq(F_109(x), F_11(x) + F_110(x))
Eq(F_110(x), F_111(x) + F_31(x) + F_67(x))
Eq(F_111(x), F_4(x)*F_70(x))
Eq(F_112(x), F_106(x) + F_113(x))
Eq(F_113(x), F_114(x) + F_125(x) + F_127(x) + F_31(x))
Eq(F_114(x), F_115(x)*F_4(x))
Eq(F_115(x), F_116(x) + F_118(x))
Eq(F_116(x), F_110(x) + F_117(x))
Eq(F_117(x), F_82(x))
Eq(F_118(x), F_113(x) + F_119(x))
Eq(F_119(x), F_120(x) + F_124(x) + 2*F_31(x) + F_90(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_122(x) + F_123(x))
Eq(F_122(x), F_117(x))
Eq(F_123(x), F_119(x))
Eq(F_124(x), 0)
Eq(F_125(x), F_126(x)*F_4(x))
Eq(F_126(x), F_106(x) + F_92(x))
Eq(F_127(x), F_128(x)*F_4(x))
Eq(F_128(x), F_46(x) + F_92(x))
Eq(F_129(x), F_130(x)*F_4(x))
Eq(F_130(x), F_106(x) + F_23(x))
Eq(F_131(x), 0)
Eq(F_132(x), F_133(x)*F_4(x))
Eq(F_133(x), F_128(x) + F_134(x))
Eq(F_134(x), F_135(x) + F_137(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_4(x)*F_46(x))
Eq(F_137(x), F_138(x))
Eq(F_138(x), F_4(x)*F_92(x))
Eq(F_139(x), F_140(x)*F_4(x))
Eq(F_140(x), F_130(x) + F_141(x))
Eq(F_141(x), F_142(x) + F_144(x))
Eq(F_142(x), F_143(x) + F_31(x) + F_47(x))
Eq(F_143(x), F_4(x)*F_61(x))
Eq(F_144(x), F_145(x) + 2*F_31(x) + F_93(x))
Eq(F_145(x), F_126(x)*F_4(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": [3, 0, 1, 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, 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": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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