0132_0213_0321_2130
Counting sequence:
1, 1, 2, 6, 20, 64, 191, 540, 1476, 3958, 10505, 27731, 72996, 191852, 503795, 1322226, 3468933, 9098392, 23858085, 62549563, 163961569, 429732177, 1126157491, 2950884404, 7731479602, 20255139898, 53060872422, 138990086697, 364054935894, 953513078684, 2497270885434, 6540128165519, 17127364561163, 44851847776696, 117451152231996, 307555015740103, 805338050782276, 2108747763808943, 5521576217482904, 14457540720543091, 37854672113380521, 99114905440331231, 259509641163244987, 679459575444066251, 1778974993262408820, 4657711610789830424, 12194732199654843706, 31927815565018097190, 83591807711280783081, 218854798423329223898, 572989304267302528050, 1500151976011236667836, 3927556966101696088831, 10282728942742996460597, 26921118100436808957070, 70481760374932976289786, 184526801617288871080773, 483104778461292314360892, 1264801793550176147766215, 3311333752164209752579378, 8669276527284556506773339, 22696674982549410786622657, 59421164900234295739776051, 155567787916387537451783599, 407284449636749210918324577, 1066290793440725276059961058, 2791600094648614431945445912, 7308537768288771996617242572, 19134078948089153039817871624, 50093851897988921429573292999, 131147832013949065182130451304, 343350470041923618784941136574, 898905498091885151617342213548, 2353370487645862681998015557241, 6161216341020034055948319178121, 16130302657093034639144575280638, 42229747706358923844166007182366, 110559070823100039696371595929693, 289447767815769190841939661857234, 757784937136558767997064334579981, 1983908681386421130261140563852966, 5193944914428372199383776014258261, 13597934913043021398626285951751147, 35599880401114848652620427166888421, 93201754124661010242736253238208045, 244005493174262651744577513436653709, 638814983910005539287086659685805646, 1672440059522960295350476479374371890, 4378506591738131615542533255358941736, 11463082963506669293440197062574176388, 30010749849036274282944692319889094395, 78569184135814134394345825984847805616, 205696843362348510122778428898739120216, 538521440808888975982309132715640049642, 1409867699581618376242688644820577607645, 3691082170576493249207422127718826310271, 9663380003892500323935814037113579262288, 25299060611571170345763937903581874790192, 66233808271382747774196946161929312698895, 173402379175066597865578140505955617381118, 453973364060632308990273220070551629791409
Generating function in Maple syntax:
-(x^7-9*x^6+29*x^5-44*x^4+44*x^3-26*x^2+8*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{7}-9 x^{6}+29 x^{5}-44 x^{4}+44 x^{3}-26 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**7 + 9*x**6 - 29*x**5 + 44*x**4 - 44*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)+x^7-9*x^6+29*x^5-44*x^4+44*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{7}-9 x^{6}+29 x^{5}-44 x^{4}+44 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
1/13800*(690*((-2/23*I+2/69*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)-69*2^(1/3)*((-37/23*I-37/69*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(-690*2^(2/3)*((-2/23*I-2/69*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+69*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)*((-37/23*I+37/69*3^(1/2))*23^(1/2)+I*3^(1/2)-1))*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*((-40*2^(2/3)*23^(1/2)*3^(1/2)+1380*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-74*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)*(23^(1/2)*3^(1/2)-69/37))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(1380*5^(1/2)+6900)*(3/2-1/2*5^(1/2))^(-n)+1/13800*(-1380*5^(1/2)+6900)*(3/2+1/2*5^(1/2))^(-n)-n^2+n
Explicit closed form in latex syntax:
\frac{\left(690 \left(\left(-\frac{2 \,\mathrm{I}}{23}+\frac{2 \sqrt{3}}{69}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-69 \,2^{\frac{1}{3}} \left(\left(-\frac{37 \,\mathrm{I}}{23}-\frac{37 \sqrt{3}}{69}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}+\frac{\left(-690 \,2^{\frac{2}{3}} \left(\left(-\frac{2 \,\mathrm{I}}{23}-\frac{2 \sqrt{3}}{69}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+69 \,2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\left(-\frac{37 \,\mathrm{I}}{23}+\frac{37 \sqrt{3}}{69}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right)\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}+\frac{\left(\left(-40 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}+1380 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-74 \,2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{69}{37}\right)\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{13800}+\frac{\left(1380 \sqrt{5}+6900\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{13800}+\frac{\left(-1380 \sqrt{5}+6900\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{13800}-n^{2}+n
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 191
a(7) = 540
a(n+5) = a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*a(n+4)-n*(n+3), n >= 8
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(5\right) = 64
a \! \left(6\right) = 191
a \! \left(7\right) = 540
a \! \left(n +5\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right)-n \left(n +3\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15255/
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[16,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[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[30,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[20,x]+F[5,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[17,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[25,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[31,x]+F[32,x]+F[80,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[49,x]
F[34,x] = F[35,x]+F[6,x]
F[35,x] = F[31,x]+F[36,x]+F[44,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[10,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[38,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[41,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[13,x]+F[46,x]
F[46,x] = F[39,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[46,x]
F[49,x] = F[30,x]+F[50,x]
F[50,x] = 2*F[31,x]+F[51,x]+F[71,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[68,x]
F[53,x] = F[54,x]+F[66,x]
F[54,x] = F[31,x]+F[55,x]+F[64,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[10,x]+F[58,x]
F[58,x] = F[15,x]+F[31,x]+F[36,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = 2*F[31,x]+F[51,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[25,x]+F[63,x]
F[63,x] = F[61,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[17,x]+F[54,x]
F[66,x] = F[67,x]
F[67,x] = F[12,x]*F[53,x]
F[68,x] = F[63,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[68,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[62,x]+F[73,x]
F[73,x] = F[74,x]+F[78,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[25,x]+F[77,x]
F[77,x] = F[75,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[73,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[65,x]+F[82,x]
F[82,x] = F[83,x]+F[93,x]
F[83,x] = F[26,x]+F[31,x]+F[84,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[57,x]+F[86,x]
F[86,x] = F[25,x]+F[87,x]
F[87,x] = 2*F[31,x]+F[29,x]+F[88,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[76,x]+F[90,x]
F[90,x] = F[28,x]+F[91,x]
F[91,x] = F[92,x]
F[92,x] = F[12,x]*F[90,x]
F[93,x] = 2*F[31,x]+F[103,x]+F[94,x]
F[94,x] = F[12,x]*F[95,x]
F[95,x] = F[96,x]+F[98,x]
F[96,x] = F[14,x]+F[97,x]
F[97,x] = F[48,x]
F[98,x] = F[63,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[101,x]*F[12,x]
F[101,x] = F[102,x]+F[77,x]
F[102,x] = F[100,x]
F[103,x] = F[12,x]*F[82,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_{16}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \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_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \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_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \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_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{80}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{49}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{30}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{51}\! \left(x \right)+F_{71}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{68}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{66}\! \left(x \right)
F_{54}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)+F_{64}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{51}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{17}\! \left(x \right)+F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{68}\! \left(x \right) = F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{62}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{25}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{75}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{65}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{93}\! \left(x \right)
F_{83}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \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_{57}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{25}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{29}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{76}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{28}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{93}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{103}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{98}\! \left(x \right)
F_{96}\! \left(x \right) = F_{14}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{48}\! \left(x \right)
F_{98}\! \left(x \right) = F_{63}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{77}\! \left(x \right)
F_{102}\! \left(x \right) = F_{100}\! \left(x \right)
F_{103}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \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_16(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_30(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_5(x))
Eq(F_20(x), F_17(x) + F_21(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_27(x))
Eq(F_24(x), F_17(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_25(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_80(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_49(x))
Eq(F_34(x), F_35(x) + F_6(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_44(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_10(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_38(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_41(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_13(x) + F_46(x))
Eq(F_46(x), F_39(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_46(x))
Eq(F_49(x), F_30(x) + F_50(x))
Eq(F_50(x), 2*F_31(x) + F_51(x) + F_71(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_68(x))
Eq(F_53(x), F_54(x) + F_66(x))
Eq(F_54(x), F_31(x) + F_55(x) + F_64(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_10(x) + F_58(x))
Eq(F_58(x), F_15(x) + F_31(x) + F_36(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), 2*F_31(x) + F_51(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_25(x) + F_63(x))
Eq(F_63(x), F_61(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_17(x) + F_54(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_12(x)*F_53(x))
Eq(F_68(x), F_63(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_68(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_62(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_78(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_25(x) + F_77(x))
Eq(F_77(x), F_75(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_73(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_65(x) + F_82(x))
Eq(F_82(x), F_83(x) + F_93(x))
Eq(F_83(x), F_26(x) + F_31(x) + F_84(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_57(x) + F_86(x))
Eq(F_86(x), F_25(x) + F_87(x))
Eq(F_87(x), F_29(x) + 2*F_31(x) + F_88(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_76(x) + F_90(x))
Eq(F_90(x), F_28(x) + F_91(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_12(x)*F_90(x))
Eq(F_93(x), F_103(x) + 2*F_31(x) + F_94(x))
Eq(F_94(x), F_12(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_98(x))
Eq(F_96(x), F_14(x) + F_97(x))
Eq(F_97(x), F_48(x))
Eq(F_98(x), F_63(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_101(x)*F_12(x))
Eq(F_101(x), F_102(x) + F_77(x))
Eq(F_102(x), F_100(x))
Eq(F_103(x), F_12(x)*F_82(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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "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], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 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": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "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], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [2, 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, 0, 2], "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": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "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], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 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": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 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