0132_1032_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 66, 212, 670, 2106, 6618, 20816, 65516, 206250, 649300, 2043998, 6434342, 20254610, 63759284, 200707520, 631806838, 1988864332, 6260744846, 19708193738, 62039404746, 195293781592, 614765103820, 1935218473202, 6091872362940, 19176599140070, 60365997952854, 190026066762338, 598182872330860, 1883019286972896, 5927554597628574, 18659343401893700, 58737729101661974, 184900440798660906, 582047919291682474, 1832228083872540224, 5767669018413774380, 18156039741328897178, 57153379993946281508, 179913069770211684590, 566348178840336791014, 1782806886044659440370, 5612096077427841170180, 17666311830417725247776, 55611765975432036920934, 175060224487902808916284, 551071911855732841761502, 1734718739934654968829482, 5460719448659324158949706, 17189793486689064587849096, 54111734340712940876931180, 170338276351443893936681570, 536207695870325817703781836, 1687927688767684529814904950, 5313425943811204559950738134, 16726128404812090467927801794, 52652163476582432721673897724, 165743694635704664043131359104, 521744416517883448475968415950, 1642398745641002234496690355124, 5170105427646063256601987531750, 16274969890179129771392773735754, 51231962023419331550259675219242, 161273043850786000143460219717808, 507671259223065135694454626949260, 1598097867363321873897701818830154, 5030650735634878088650912204125236, 15835980599672611868605006576894334, 49850068059149135082364991514031782, 156922981173213582812295841349700178, 493977701115079537870693363073861332, 1554991928999826538128250885400638208, 4894957593827281121713219293730669014, 15408832289424724410338817950514099500, 48505448305216880814343864943703982638, 152690253946420631148072225354443254538, 480653503159456606559596124537175169738, 1513048699103781413608511686754196991032, 4762924540883176465903330598209949895564, 14993205569380818229347383456564627965010, 47197097353977524933288053718382859164380, 148571697248650821511013395103147014285830, 467688702502051529473536802716203130870742, 1472236815616231178294395079275953363045410, 4634452852207420946267983304058351073855244, 14588789664483041779662284781117349290644384, 45924036916926135458139584291561501211035006, 144564231526462837439839466126943511953039908, 455073605019555985164601396414206893469698998, 1432525762415761563797026771567576131416755690, 4509446466130841137134142363154662541583054506, 14195282182295635842310090534330163849923738784, 44685315093205781521634961819424243906362547308, 140664860292067059696743718037861076248517849722, 442798778070946530064380077227501596387419140420, 1393885846500790722688924116615847259972997204494, 4387811912082390360259299952818745326978249246438, 13812388886898135803034807538654941405856401223346
Generating function in Maple syntax:
-(x-1)*(3*x^3-5*x^2+4*x-1)/(2*x^2-2*x+1)/(x^3-3*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}{\left(2 x^{2}-2 x +1\right) \left(x^{3}-3 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(1 - x)*(3*x**3 - 5*x**2 + 4*x - 1)/((2*x**2 - 2*x + 1)*(x**3 - 3*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x^2-2*x+1)*(x^3-3*x^2+4*x-1)*F(x)+(x-1)*(3*x^3-5*x^2+4*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}-2 x +1\right) \left(x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) = 0
Explicit closed form in Maple syntax:
5/1116*(((I*3^(1/3)+1/3*3^(5/6))*31^(1/2)-31/5*I*3^(5/6)-31/5*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)+248/5-8/5*((I*3^(2/3)-3^(1/6))*31^(1/2)-31/4*I*3^(1/6)+31/12*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3))*((53/2232*((I*3^(2/3)+97/53*3^(1/6))*31^(1/2)-341/53*I*3^(1/6)-961/159*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3)-17/558*((I*3^(1/3)+11/51*3^(5/6))*31^(1/2)-155/34*I*3^(5/6)+31/34*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)-3/62*I*31^(1/2)+7/18)*(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)+(-11/1116*((I*3^(2/3)+64/11*3^(1/6))*31^(1/2)-155/11*I*3^(1/6)-496/33*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3)-7/279*2^(2/3)*((I*3^(1/3)+10/21*3^(5/6))*31^(1/2)-31/14*I*3^(5/6)-62/7*3^(1/3))*(9+31^(1/2)*3^(1/2))^(1/3)+3/62*I*31^(1/2)+7/18)*(-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)+1/288*((1/2-1/2*I)^(-n)+(1/2+1/2*I)^(-n))*((I*3^(2/3)-3^(1/6))*31^(1/2)-I*3^(1/6)+1/3*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3)-1/72*((I*3^(1/3)+1/3*3^(5/6))*31^(1/2)-7/2*3^(1/3)-7/2*I*3^(5/6))*2^(2/3)*((1/2-1/2*I)^(-n)+(1/2+1/2*I)^(-n))*(9+31^(1/2)*3^(1/2))^(1/3)+4/9*(1/2-1/2*I)^(-n)+4/9*(1/2+1/2*I)^(-n)+(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))
Explicit closed form in latex syntax:
\frac{5 \left(\left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{5}{6}}}{5}-\frac{31 \,3^{\frac{1}{3}}}{5}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{248}{5}-\frac{8 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{1}{6}}}{4}+\frac{31 \,3^{\frac{2}{3}}}{12}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{5}\right) \left(\left(\frac{53 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+\frac{97 \,3^{\frac{1}{6}}}{53}\right) \sqrt{31}-\frac{341 \,\mathrm{I} \,3^{\frac{1}{6}}}{53}-\frac{961 \,3^{\frac{2}{3}}}{159}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{2232}-\frac{17 \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{11 \,3^{\frac{5}{6}}}{51}\right) \sqrt{31}-\frac{155 \,\mathrm{I} \,3^{\frac{5}{6}}}{34}+\frac{31 \,3^{\frac{1}{3}}}{34}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{558}-\frac{3 \,\mathrm{I} \sqrt{31}}{62}+\frac{7}{18}\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}+\left(-\frac{11 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+\frac{64 \,3^{\frac{1}{6}}}{11}\right) \sqrt{31}-\frac{155 \,\mathrm{I} \,3^{\frac{1}{6}}}{11}-\frac{496 \,3^{\frac{2}{3}}}{33}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{1116}-\frac{7 \,2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{10 \,3^{\frac{5}{6}}}{21}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{5}{6}}}{14}-\frac{62 \,3^{\frac{1}{3}}}{7}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{279}+\frac{3 \,\mathrm{I} \sqrt{31}}{62}+\frac{7}{18}\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}+\frac{\left(\left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)^{-n}+\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)^{-n}\right) \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{31}-\mathrm{I} \,3^{\frac{1}{6}}+\frac{3^{\frac{2}{3}}}{3}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{288}-\frac{\left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{7 \,3^{\frac{1}{3}}}{2}-\frac{7 \,\mathrm{I} \,3^{\frac{5}{6}}}{2}\right) 2^{\frac{2}{3}} \left(\left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)^{-n}+\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)^{-n}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{72}+\frac{4 \left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)^{-n}}{9}+\frac{4 \left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)^{-n}}{9}+\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}\right)}{1116}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = 2*a(n)-8*a(n+1)+15*a(n+2)-13*a(n+3)+6*a(n+4), 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 +5\right) = 2 a \! \left(n \right)-8 a \! \left(n +1\right)+15 a \! \left(n +2\right)-13 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15523/
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[28,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[7,x]
F[14,x] = F[15,x]+F[16,x]+F[27,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[22,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[21,x]*F[4,x]
F[27,x] = F[13,x]*F[4,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[63,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[11,x]+F[14,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[34,x]
F[37,x] = 2*F[15,x]+F[38,x]+F[60,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[44,x]+F[61,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[44,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[36,x]+F[50,x]
F[50,x] = F[47,x]+F[51,x]
F[51,x] = 2*F[15,x]+F[52,x]+F[60,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[34,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[54,x]
F[57,x] = F[44,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[57,x]
F[60,x] = F[4,x]*F[50,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[43,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[2,x]+F[66,x]
F[66,x] = F[15,x]+F[30,x]+F[67,x]
F[67,x] = F[4,x]*F[65,x]
F[68,x] = F[47,x]+F[69,x]
F[69,x] = 2*F[15,x]+F[52,x]+F[70,x]
F[70,x] = F[4,x]*F[68,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_{28}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{63}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{60}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{61}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{36}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{34}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{44}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \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_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{2}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{47}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \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_28(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_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_27(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_22(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_21(x)*F_4(x))
Eq(F_27(x), F_13(x)*F_4(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_63(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_11(x) + F_14(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_34(x))
Eq(F_37(x), 2*F_15(x) + F_38(x) + F_60(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_44(x) + F_61(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_44(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_36(x) + F_50(x))
Eq(F_50(x), F_47(x) + F_51(x))
Eq(F_51(x), 2*F_15(x) + F_52(x) + F_60(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_34(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_54(x))
Eq(F_57(x), F_44(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_57(x))
Eq(F_60(x), F_4(x)*F_50(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_43(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_2(x) + F_66(x))
Eq(F_66(x), F_15(x) + F_30(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_65(x))
Eq(F_68(x), F_47(x) + F_69(x))
Eq(F_69(x), 2*F_15(x) + F_52(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_68(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": [1, 0, 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, 0, 3], "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": [1, 0, 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, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"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, 1], [0, 0]]}, {"patt": [1, 0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 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"pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"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, 1], [0, 0]]}, {"patt": [1, 0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, 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"pos": [[0, 1], [0, 0], [0, 1]]}, {"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, 1], [0, 0]]}, {"patt": [1, 0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 1]]}], "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, 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