0123_0132_1203_3021
Counting sequence:
1, 1, 2, 6, 20, 63, 187, 527, 1424, 3725, 9508, 23822, 58841, 143739, 348083, 837088, 2001815, 4765336, 11301464, 26719563, 63009187, 148266115, 348252004, 816737949, 1912985956, 4475754022, 10462068245, 24435762979, 57035067183, 133048065316, 310214467575, 722991625360, 1684412578472, 3923106096783, 9134759047851, 21265026106695, 49493763477704, 115176191580717, 267986462519796, 623461545128446, 1450309456432113, 3373434850570619, 7846033401998907, 18247312397333336, 42434850110664375, 98679048683508568, 229460937714896792, 533551924865070707, 1240597665802393379, 2884519521298868155, 6706654028970753788, 15592998453744649277, 36253162311846770196, 84286055031532689494, 195956660873671024813, 455574678767838160643, 1059144057383792511895, 2462334051081547019516, 5724477868429243756119, 13308287862877128862768, 30938998507591844338792, 71926410874925315533495, 167212549891418979758859, 388729879269897338494879, 903703054577577743393536, 2100886166435954276736077, 4884030692125798169488900, 11354119643489450673533934, 26395377403465483908651913, 61362310997168896448960571, 142651072591316268004279555, 331625522709578686625905040, 770939832996276803522787927, 1792225724266998071730176088, 4166435425730036827589543256, 9685819454076426208468437275, 22516862819884273173231145059, 52345484146926144192895591347, 121688744594478592965791896276, 282892524988252324735013601501, 657646363280295054566748504068, 1528844371173997351396802007174, 3554136085379930758193506218437, 8262372223932645603156830352291, 19207701565933178598086615041855, 44652521722756387253352359468948, 103804585035219873981466016334647, 241316514775848873664630620426512, 560993099079400955542013538716584, 1304151258920800383793648092906335, 3031784905777598894992353224262443, 7048046923366898434993387830791607, 16384725467058901022912326746957176, 38089873959405711341989001023952749, 88548230865836642911628623849539668, 205849696142498232066805969129279198, 478542552648708771705121206475928225, 1112476600513929793865323396545200571, 2586194600360796670196419928865893899, 6012173568651092394451506566088947848, 13976608937641320036602547969291734647
Generating function in Maple syntax:
-(x^7-3*x^6+4*x^5-10*x^4+15*x^3-14*x^2+6*x-1)/(2*x-1)/(x^3-2*x^2+3*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{7}-3 x^{6}+4 x^{5}-10 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**7 + 3*x**6 - 4*x**5 + 10*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**2*(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3-2*x^2+3*x-1)*(x-1)^2*F(x)+x^7-3*x^6+4*x^5-10*x^4+15*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}-3 x^{6}+4 x^{5}-10 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/2760*(-115*2^(1/3)*((I-1/23*23^(1/2))*3^(1/2)-3/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+4600-23*2^(2/3)*((I+29/23*23^(1/2))*3^(1/2)-87/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(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)+1/2760*(23*((I-29/23*23^(1/2))*3^(1/2)-87/23*I*23^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+115*2^(1/3)*((I+1/23*23^(1/2))*3^(1/2)-3/23*I*23^(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)+1/2760*((58*2^(2/3)*23^(1/2)*3^(1/2)-46*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+(-10*2^(1/3)*3^(1/2)*23^(1/2)+230*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)+2*n-21/4*2^n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-115 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{\sqrt{23}}{23}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+4600-23 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{29 \sqrt{23}}{23}\right) \sqrt{3}-\frac{87 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\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}}{2760}\\+\\\frac{\left(23 \left(\left(\mathrm{I}-\frac{29 \sqrt{23}}{23}\right) \sqrt{3}-\frac{87 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+115 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{\sqrt{23}}{23}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{23}}{23}-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}}{2760}\\+\\\frac{\left(\left(58 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-46 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+\left(-10 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{23}+230 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{2760}\\+2 n -\frac{21 \,2^{n}}{4}+2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 187
a(7) = 527
a(n+4) = -2*a(n)+5*a(n+1)-8*a(n+2)+5*a(n+3)+2*n+2, 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) = 63
a \! \left(6\right) = 187
a \! \left(7\right) = 527
a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+2 n +2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14840/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[20,x]+F[45,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[13,x]+F[28,x]+F[43,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[16,x]*F[4,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = 2*F[13,x]+F[35,x]+F[39,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[31,x]
F[38,x] = F[34,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[20,x]*F[4,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[20,x]+F[41,x]
F[45,x] = F[113,x]+F[13,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[62,x]
F[48,x] = F[49,x]+F[7,x]
F[49,x] = F[13,x]+F[32,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[24,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[24,x]*F[4,x]
F[55,x] = F[49,x]+F[56,x]
F[56,x] = 2*F[13,x]+F[57,x]+F[61,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[53,x]
F[60,x] = F[56,x]
F[61,x] = F[4,x]*F[49,x]
F[62,x] = F[45,x]+F[63,x]
F[63,x] = F[100,x]+F[13,x]+F[39,x]+F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[80,x]
F[66,x] = F[49,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[31,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[31,x]*F[4,x]
F[73,x] = F[67,x]+F[74,x]
F[74,x] = 3*F[13,x]+F[75,x]+F[79,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[78,x]
F[77,x] = F[71,x]
F[78,x] = F[74,x]
F[79,x] = F[4,x]*F[67,x]
F[80,x] = F[63,x]+F[81,x]
F[81,x] = 2*F[13,x]+F[82,x]+F[86,x]+F[87,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[67,x]
F[85,x] = F[81,x]
F[86,x] = 0
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[92,x]
F[89,x] = F[34,x]+F[90,x]
F[90,x] = F[91,x]
F[91,x] = F[34,x]*F[4,x]
F[92,x] = F[93,x]+F[94,x]
F[93,x] = F[87,x]
F[94,x] = 4*F[13,x]+F[95,x]+F[99,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[97,x]+F[98,x]
F[97,x] = F[90,x]
F[98,x] = F[94,x]
F[99,x] = F[4,x]*F[93,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[105,x]
F[102,x] = F[103,x]+F[27,x]
F[103,x] = F[104,x]
F[104,x] = F[27,x]*F[4,x]
F[105,x] = F[106,x]+F[107,x]
F[106,x] = F[100,x]+F[13,x]+F[35,x]+F[39,x]
F[107,x] = 3*F[13,x]+F[108,x]+F[112,x]
F[108,x] = F[109,x]*F[4,x]
F[109,x] = F[110,x]+F[111,x]
F[110,x] = F[103,x]
F[111,x] = F[107,x]
F[112,x] = F[106,x]*F[4,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[44,x]
F[115,x] = F[116,x]+F[117,x]
F[116,x] = F[113,x]+F[13,x]+F[28,x]
F[117,x] = 2*F[13,x]+F[118,x]+F[121,x]
F[118,x] = F[119,x]*F[4,x]
F[119,x] = F[120,x]+F[40,x]
F[120,x] = F[117,x]
F[121,x] = F[116,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{45}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)+F_{43}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{113}\! \left(x \right)+F_{13}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{62}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{7}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{24}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{53}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{62}\! \left(x \right) = F_{45}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{100}\! \left(x \right)+F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{80}\! \left(x \right)
F_{66}\! \left(x \right) = F_{49}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \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_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{31}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 3 F_{13}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{71}\! \left(x \right)
F_{78}\! \left(x \right) = F_{74}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{80}\! \left(x \right) = F_{63}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{67}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = 0
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{92}\! \left(x \right)
F_{89}\! \left(x \right) = F_{34}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{94}\! \left(x \right)
F_{93}\! \left(x \right) = F_{87}\! \left(x \right)
F_{94}\! \left(x \right) = 4 F_{13}\! \left(x \right)+F_{95}\! \left(x \right)+F_{99}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)+F_{98}\! \left(x \right)
F_{97}\! \left(x \right) = F_{90}\! \left(x \right)
F_{98}\! \left(x \right) = F_{94}\! \left(x \right)
F_{99}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \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_{105}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{27}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{107}\! \left(x \right)
F_{106}\! \left(x \right) = F_{100}\! \left(x \right)+F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{107}\! \left(x \right) = 3 F_{13}\! \left(x \right)+F_{108}\! \left(x \right)+F_{112}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{4}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{111}\! \left(x \right)
F_{110}\! \left(x \right) = F_{103}\! \left(x \right)
F_{111}\! \left(x \right) = F_{107}\! \left(x \right)
F_{112}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{44}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{117}\! \left(x \right)
F_{116}\! \left(x \right) = F_{113}\! \left(x \right)+F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{117}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{118}\! \left(x \right)+F_{121}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{4}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{40}\! \left(x \right)
F_{120}\! \left(x \right) = F_{117}\! \left(x \right)
F_{121}\! \left(x \right) = F_{116}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_20(x) + F_45(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_13(x) + F_28(x) + F_43(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_16(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), 2*F_13(x) + F_35(x) + F_39(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_31(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_20(x)*F_4(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_20(x) + F_41(x))
Eq(F_45(x), F_113(x) + F_13(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_62(x))
Eq(F_48(x), F_49(x) + F_7(x))
Eq(F_49(x), F_13(x) + F_32(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_24(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_24(x)*F_4(x))
Eq(F_55(x), F_49(x) + F_56(x))
Eq(F_56(x), 2*F_13(x) + F_57(x) + F_61(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_53(x))
Eq(F_60(x), F_56(x))
Eq(F_61(x), F_4(x)*F_49(x))
Eq(F_62(x), F_45(x) + F_63(x))
Eq(F_63(x), F_100(x) + F_13(x) + F_39(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_80(x))
Eq(F_66(x), F_49(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_31(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_31(x)*F_4(x))
Eq(F_73(x), F_67(x) + F_74(x))
Eq(F_74(x), 3*F_13(x) + F_75(x) + F_79(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_78(x))
Eq(F_77(x), F_71(x))
Eq(F_78(x), F_74(x))
Eq(F_79(x), F_4(x)*F_67(x))
Eq(F_80(x), F_63(x) + F_81(x))
Eq(F_81(x), 2*F_13(x) + F_82(x) + F_86(x) + F_87(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_67(x))
Eq(F_85(x), F_81(x))
Eq(F_86(x), 0)
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_92(x))
Eq(F_89(x), F_34(x) + F_90(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_34(x)*F_4(x))
Eq(F_92(x), F_93(x) + F_94(x))
Eq(F_93(x), F_87(x))
Eq(F_94(x), 4*F_13(x) + F_95(x) + F_99(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_97(x) + F_98(x))
Eq(F_97(x), F_90(x))
Eq(F_98(x), F_94(x))
Eq(F_99(x), F_4(x)*F_93(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_105(x))
Eq(F_102(x), F_103(x) + F_27(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_27(x)*F_4(x))
Eq(F_105(x), F_106(x) + F_107(x))
Eq(F_106(x), F_100(x) + F_13(x) + F_35(x) + F_39(x))
Eq(F_107(x), F_108(x) + F_112(x) + 3*F_13(x))
Eq(F_108(x), F_109(x)*F_4(x))
Eq(F_109(x), F_110(x) + F_111(x))
Eq(F_110(x), F_103(x))
Eq(F_111(x), F_107(x))
Eq(F_112(x), F_106(x)*F_4(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_44(x))
Eq(F_115(x), F_116(x) + F_117(x))
Eq(F_116(x), F_113(x) + F_13(x) + F_28(x))
Eq(F_117(x), F_118(x) + F_121(x) + 2*F_13(x))
Eq(F_118(x), F_119(x)*F_4(x))
Eq(F_119(x), F_120(x) + F_40(x))
Eq(F_120(x), F_117(x))
Eq(F_121(x), F_116(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [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, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": 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