0132_0231_1320_2130
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 606, 1774, 5107, 14534, 41034, 115208, 322193, 898546, 2500938, 6951034, 19299859, 53547782, 148491126, 411618668, 1140699089, 3160542586, 8755693062, 24253527142, 67178013859, 186061373318, 515309964882, 1427147198480, 3952394330465, 10945748410978, 30312803407314, 83946666754162, 232476166491619, 643801078578950, 1782886620901278, 4937360720420276, 13673048158546049, 37864773455334250, 104858839086307182, 290385204278993758, 804162371839729171, 2226962399593942598, 6167113329755975706, 17078546558583480344, 47295501939727941233, 130975098978635385682, 362708394331319829402, 1004445691508168310250, 2781603990278053020019, 7703075095869775645190, 21332067833264372357766, 59074734083051254712060, 163595212889458887951473, 453042979173989222555866, 1254608472226358916539670, 3474377677416144179237590, 9621567593239258372174915, 26644933682614380910718726, 73787611348192810653002658, 204339468387548873648031776, 565875728361759319656946625, 1567075329775304152886081218, 4339689732810466526469869730, 12017869603935383773821556450, 33280994422606047003163044547, 92164803424363339872701774342, 255231285509759588609892489774, 706810047638201937402396049220, 1957363661050570492354512051617, 5420512221465990986040061834186, 15010983051871048118706127977726, 41569800596968611293508951105742, 115118930964042551610013021083955, 318797975354113641379540707944390, 882844795709434083270726989496042, 2444855342766565358939871152909864, 6770519208001044032722862919742289, 18749547077059337618088316080941746, 51922977366170207771659699446250218, 143789904229905676911969131118066010, 398196282401512854947046981869919187, 1102721920342616484430253457700001030, 3053759382859869052714526409872268438, 8456752510645757083578060528834960524, 23419220069434525792285284184429088273, 64854667080546703565663069534517520954, 179601533682909370427623487148344331558, 497369154037731990388175680495716354886, 1377360595511065418059932632674865839843, 3814314166178837861733977722280312667782, 10562951057064108612559640249867962861554, 29251899600526410304063803730410790443056, 81007061910697673981483604804469418508065, 224332237188638147893182363214514822688034, 621241549255758035282598713132784629489778, 1720399472489363342497040230114906063270738, 4764289105401029561223590038679643540973219, 13193709392969602218700099995563199505064198, 36537238545997378121861292971137641669341118, 101182295350424037077082454542554930307634388
Generating function in Maple syntax:
-(3*x^4-4*x^3+8*x^2-5*x+1)/(-1+x)/(2*x-1)/(x^3-x^2+3*x-1)
Generating function in latex syntax:
-\frac{3 x^{4}-4 x^{3}+8 x^{2}-5 x +1}{\left(-1+x \right) \left(2 x -1\right) \left(x^{3}-x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(-3*x**4 + 4*x**3 - 8*x**2 + 5*x - 1)/((x - 1)*(2*x - 1)*(x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(-1+x)*(2*x-1)*(x^3-x^2+3*x-1)*F(x)+3*x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(-1+x \right) \left(2 x -1\right) \left(x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+3 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
1/7296*(((-108*I-36*3^(1/2))*19^(1/2)+76*I*3^(1/2)+76)*(1+3*19^(1/2)*3^(1/2))^(1/3)+1216+((9*I-3*3^(1/2))*19^(1/2)+95*I*3^(1/2)-95)*(1+3*19^(1/2)*3^(1/2))^(2/3))*(1/384*((I+3*19^(1/2))*3^(1/2)-9*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-1/6*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3)^(-n)+1/7296*(((108*I-36*3^(1/2))*19^(1/2)-76*I*3^(1/2)+76)*(1+3*19^(1/2)*3^(1/2))^(1/3)+1216+((-9*I-3*3^(1/2))*19^(1/2)-95*I*3^(1/2)-95)*(1+3*19^(1/2)*3^(1/2))^(2/3))*(1/384*((-I+3*19^(1/2))*3^(1/2)+9*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-1/6*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3)^(-n)+1/7296*((72*19^(1/2)*3^(1/2)-152)*(1+3*19^(1/2)*3^(1/2))^(1/3)+6*(1+3*19^(1/2)*3^(1/2))^(2/3)*19^(1/2)*3^(1/2)+190*(1+3*19^(1/2)*3^(1/2))^(2/3)+1216)*(1/3*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3+1/192*(1+3*19^(1/2)*3^(1/2))^(2/3)-1/64*(1+3*19^(1/2)*3^(1/2))^(2/3)*19^(1/2)*3^(1/2))^(-n)-2^n+3/2
Explicit closed form in latex syntax:
\frac{\left(\left(\left(-108 \,\mathrm{I}-36 \sqrt{3}\right) \sqrt{19}+76 \,\mathrm{I} \sqrt{3}+76\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+1216+\left(\left(9 \,\mathrm{I}-3 \sqrt{3}\right) \sqrt{19}+95 \,\mathrm{I} \sqrt{3}-95\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{7296}+\frac{\left(\left(\left(108 \,\mathrm{I}-36 \sqrt{3}\right) \sqrt{19}-76 \,\mathrm{I} \sqrt{3}+76\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+1216+\left(\left(-9 \,\mathrm{I}-3 \sqrt{3}\right) \sqrt{19}-95 \,\mathrm{I} \sqrt{3}-95\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{7296}+\frac{\left(\left(72 \sqrt{19}\, \sqrt{3}-152\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}+190 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+1216\right) \left(\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}}{64}\right)^{-n}}{7296}-2^{n}+\frac{3}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+4) = -2*a(n)+3*a(n+1)-7*a(n+2)+5*a(n+3)+3, 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 +4\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)-7 a \! \left(n +2\right)+5 a \! \left(n +3\right)+3, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15365/
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[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[9,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[18,x]+F[30,x]+F[81,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[51,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[18,x]+F[34,x]+F[37,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[34,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[43,x]
F[39,x] = F[12,x]+F[40,x]
F[40,x] = F[18,x]+F[41,x]+F[42,x]
F[41,x] = F[10,x]*F[12,x]
F[42,x] = F[12,x]*F[39,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[12,x]+F[44,x]
F[47,x] = 2*F[18,x]+F[48,x]+F[50,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[40,x]+F[47,x]
F[50,x] = F[12,x]*F[43,x]
F[51,x] = F[29,x]+F[52,x]
F[52,x] = 2*F[18,x]+F[53,x]+F[71,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[70,x]
F[55,x] = F[18,x]+F[56,x]+F[68,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[60,x]
F[58,x] = F[10,x]+F[59,x]
F[59,x] = F[18,x]+F[34,x]+F[42,x]
F[60,x] = F[55,x]+F[61,x]
F[61,x] = 2*F[18,x]+F[53,x]+F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[23,x]
F[66,x] = 2*F[18,x]+F[62,x]+F[67,x]
F[67,x] = F[12,x]*F[55,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[23,x]+F[55,x]
F[70,x] = F[53,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[63,x]+F[73,x]
F[73,x] = F[74,x]+F[77,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[64,x]+F[74,x]
F[77,x] = 3*F[18,x]+F[78,x]+F[80,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[66,x]+F[77,x]
F[80,x] = F[12,x]*F[73,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[69,x]+F[83,x]
F[83,x] = F[84,x]+F[95,x]
F[84,x] = F[18,x]+F[85,x]+F[93,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[87,x]+F[90,x]
F[87,x] = F[12,x]+F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[12,x]*F[87,x]
F[90,x] = F[64,x]+F[91,x]
F[91,x] = F[92,x]
F[92,x] = F[12,x]*F[90,x]
F[93,x] = F[12,x]*F[94,x]
F[94,x] = F[23,x]+F[84,x]
F[95,x] = F[110,x]+F[112,x]+F[18,x]+F[96,x]
F[96,x] = F[12,x]*F[97,x]
F[97,x] = F[104,x]+F[98,x]
F[98,x] = F[40,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[103,x]
F[102,x] = F[41,x]
F[103,x] = F[100,x]
F[104,x] = F[105,x]+F[66,x]
F[105,x] = F[106,x]
F[106,x] = F[107,x]*F[12,x]
F[107,x] = F[108,x]+F[109,x]
F[108,x] = F[67,x]
F[109,x] = F[106,x]
F[110,x] = F[111,x]*F[12,x]
F[111,x] = F[55,x]+F[95,x]
F[112,x] = F[12,x]*F[83,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_{22}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{81}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{51}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{51}\! \left(x \right) = F_{29}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{71}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{70}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{23}\! \left(x \right)+F_{55}\! \left(x \right)
F_{70}\! \left(x \right) = F_{53}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{63}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{77}\! \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_{64}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{66}\! \left(x \right)+F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{69}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{95}\! \left(x \right)
F_{84}\! \left(x \right) = F_{18}\! \left(x \right)+F_{85}\! \left(x \right)+F_{93}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{90}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{64}\! \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) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{23}\! \left(x \right)+F_{84}\! \left(x \right)
F_{95}\! \left(x \right) = F_{110}\! \left(x \right)+F_{112}\! \left(x \right)+F_{18}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{12}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{104}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{40}\! \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_{103}\! \left(x \right)
F_{102}\! \left(x \right) = F_{41}\! \left(x \right)
F_{103}\! \left(x \right) = F_{100}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{66}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{12}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{109}\! \left(x \right)
F_{108}\! \left(x \right) = F_{67}\! \left(x \right)
F_{109}\! \left(x \right) = F_{106}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{12}\! \left(x \right)
F_{111}\! \left(x \right) = F_{55}\! \left(x \right)+F_{95}\! \left(x \right)
F_{112}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \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_22(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_9(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_18(x) + F_30(x) + F_81(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_51(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_18(x) + F_34(x) + F_37(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_34(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_43(x))
Eq(F_39(x), F_12(x) + F_40(x))
Eq(F_40(x), F_18(x) + F_41(x) + F_42(x))
Eq(F_41(x), F_10(x)*F_12(x))
Eq(F_42(x), F_12(x)*F_39(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_12(x) + F_44(x))
Eq(F_47(x), 2*F_18(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_40(x) + F_47(x))
Eq(F_50(x), F_12(x)*F_43(x))
Eq(F_51(x), F_29(x) + F_52(x))
Eq(F_52(x), 2*F_18(x) + F_53(x) + F_71(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_70(x))
Eq(F_55(x), F_18(x) + F_56(x) + F_68(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_60(x))
Eq(F_58(x), F_10(x) + F_59(x))
Eq(F_59(x), F_18(x) + F_34(x) + F_42(x))
Eq(F_60(x), F_55(x) + F_61(x))
Eq(F_61(x), 2*F_18(x) + F_53(x) + F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_23(x))
Eq(F_66(x), 2*F_18(x) + F_62(x) + F_67(x))
Eq(F_67(x), F_12(x)*F_55(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_23(x) + F_55(x))
Eq(F_70(x), F_53(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_63(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_77(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_64(x) + F_74(x))
Eq(F_77(x), 3*F_18(x) + F_78(x) + F_80(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_66(x) + F_77(x))
Eq(F_80(x), F_12(x)*F_73(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_69(x) + F_83(x))
Eq(F_83(x), F_84(x) + F_95(x))
Eq(F_84(x), F_18(x) + F_85(x) + F_93(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_90(x))
Eq(F_87(x), F_12(x) + F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_12(x)*F_87(x))
Eq(F_90(x), F_64(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_12(x)*F_94(x))
Eq(F_94(x), F_23(x) + F_84(x))
Eq(F_95(x), F_110(x) + F_112(x) + F_18(x) + F_96(x))
Eq(F_96(x), F_12(x)*F_97(x))
Eq(F_97(x), F_104(x) + F_98(x))
Eq(F_98(x), F_40(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_103(x))
Eq(F_102(x), F_41(x))
Eq(F_103(x), F_100(x))
Eq(F_104(x), F_105(x) + F_66(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_107(x)*F_12(x))
Eq(F_107(x), F_108(x) + F_109(x))
Eq(F_108(x), F_67(x))
Eq(F_109(x), F_106(x))
Eq(F_110(x), F_111(x)*F_12(x))
Eq(F_111(x), F_55(x) + F_95(x))
Eq(F_112(x), F_12(x)*F_83(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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [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, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": 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