0231_1032_1203_1230_1302
Counting sequence:
1, 1, 2, 6, 19, 57, 165, 468, 1309, 3622, 9936, 27067, 73313, 197633, 530654, 1420059, 3789338, 10087008, 26794945, 71049313, 188099987, 497311914, 1313277053, 3464467600, 9131149482, 24047596639, 63287388699, 166455606399, 437569660586, 1149717740005, 3019637230132, 7927916463296, 20807636721711, 54596176032595, 143216111757343, 375598980283680, 984852033794655, 2581912313495936, 6767754895522436, 17737323159468877, 46481342156813795, 121793014448795581, 319098350241580472, 835968488730251861, 2189891486345901214, 5736226825641025562, 15024649268238750635, 39351344471789530373, 103061054925283176029, 269905445930752971452, 706826441683462292481, 1850971775093860956522, 4847013879514521594108, 12692220218066939329963, 33234645742679427924749, 87023338200632669039889, 227862384849833525427958, 596626620907383228280863, 1562163480757293554654790, 4090203236893013925563268, 10709235275295858222230857, 28039336896944835157197469, 73413039664169766987054123, 192209695270931386185102130, 503239091485413187652346977, 1317561153093571843999641044, 3449568912021882674242630694, 9031435113172078630045104879, 23645409503549463577278838631, 61906358109468218613212221935, 162077302338920693525164804402, 424334005101561663089247655801, 1110944371232428603109281413056, 2908544808521246967347582381700, 7614796293768819948653982091463, 19936091049513354492924644624335, 52194051006006175894718747055975, 136647396708191194706127937263016, 357751242011883749653466777074131, 936613542679271967878483644818132, 2452106155034421233509910825993840, 6419743905639151485416408049627661, 16807216186863346235603148885913567, 44002115332469995833982153532871341, 115199619576358500831148144577390784, 301597881963983181141562384506879817, 789596673163619455522818771061362730, 2067198290702016934815337839281083278, 5412012503339177445712067058664862499, 14168872473003498606839872213272794657, 37094682221116917544175275497669316901, 97115353903704831221539986588046936852, 254251797272867092573790338234934535653, 665641009142235440637527651322038029918, 1742673487983474756041045209718409330344, 4562384703625286935770318167352779612219, 11944492824911748316857231296129520266953, 31271122137363484121696173947352603944593, 81868939530717655946025751406819167964510, 214335849754918649461562188406933474390723, 561138966113601411985510014720002048493602
Generating function in Maple syntax:
(x^5-2*x^4+4*x^3-8*x^2+5*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
\frac{x^{5}-2 x^{4}+4 x^{3}-8 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(x**5 - 2*x**4 + 4*x**3 - 8*x**2 + 5*x - 1)/((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)*F(x)-x^5+2*x^4-4*x^3+8*x^2-5*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) F \! \left(x \right)-x^{5}+2 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/40*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)*(((-I+7/69*23^(1/2))*3^(1/2)+7/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*((I-13/69*23^(1/2))*3^(1/2)+13/23*I*23^(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)+1/40*(((-I-7/69*23^(1/2))*3^(1/2)+7/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*((I+13/69*23^(1/2))*3^(1/2)+13/23*I*23^(1/2)+1))*2^(1/3)*(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*((-26*3^(1/2)*2^(2/3)*23^(1/2)-138*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+14*2^(1/3)*(23^(1/2)*3^(1/2)-69/7)*(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)+2/5*5^(1/2)*((3/2-1/2*5^(1/2))^(-n)-(3/2+1/2*5^(1/2))^(-n)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\left(\left(-\mathrm{I}+\frac{7 \sqrt{23}}{69}\right) \sqrt{3}+\frac{7 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{13 \sqrt{23}}{69}\right) \sqrt{3}+\frac{13 \,\mathrm{I} \sqrt{23}}{23}-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}}{40}\\+\\\frac{\left(\left(\left(-\mathrm{I}-\frac{7 \sqrt{23}}{69}\right) \sqrt{3}+\frac{7 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{13 \sqrt{23}}{69}\right) \sqrt{3}+\frac{13 \,\mathrm{I} \sqrt{23}}{23}+1\right)\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} \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}}{40}\\+\\\frac{\left(\left(-26 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-138 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+14 \,2^{\frac{1}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{69}{7}\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}\\+\frac{2 \sqrt{5}\, \left(\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}-\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}\right)}{5} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(n+5) = a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*a(n+4), n >= 6
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) = 19
a \! \left(5\right) = 57
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), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22013/
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[13,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[13,x]*F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]*F[7,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[11,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[5,x]
F[16,x] = F[17,x]+F[72,x]
F[17,x] = F[18,x]+F[41,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[22,x]
F[25,x] = F[19,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[39,x]
F[27,x] = 0
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[13,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[23,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[27,x]+F[28,x]+F[35,x]
F[35,x] = F[13,x]*F[19,x]
F[36,x] = F[37,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[13,x]*F[40,x]
F[40,x] = F[19,x]+F[38,x]
F[41,x] = F[23,x]+F[42,x]
F[42,x] = F[27,x]+F[43,x]+F[45,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[23,x]
F[45,x] = F[13,x]*F[46,x]
F[46,x] = F[47,x]+F[54,x]
F[47,x] = F[23,x]+F[48,x]
F[48,x] = F[27,x]+F[49,x]+F[51,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = F[13,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[23,x]+F[53,x]
F[53,x] = F[51,x]
F[54,x] = F[42,x]+F[55,x]
F[55,x] = F[27,x]+F[56,x]+F[58,x]+F[70,x]
F[56,x] = F[13,x]*F[57,x]
F[57,x] = F[31,x]
F[58,x] = F[13,x]*F[59,x]
F[59,x] = F[60,x]+F[64,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[27,x]+F[32,x]+F[49,x]
F[62,x] = F[63,x]
F[63,x] = F[13,x]*F[53,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = F[27,x]+F[56,x]+F[58,x]+F[66,x]
F[66,x] = F[13,x]*F[42,x]
F[67,x] = F[68,x]
F[68,x] = F[13,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[13,x]*F[71,x]
F[71,x] = F[42,x]+F[69,x]
F[72,x] = F[111,x]+F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[12,x]
F[75,x] = F[27,x]+F[76,x]+F[81,x]
F[76,x] = F[13,x]*F[77,x]
F[77,x] = F[23,x]+F[78,x]
F[78,x] = F[27,x]+F[76,x]+F[79,x]
F[79,x] = F[13,x]*F[80,x]
F[80,x] = F[74,x]+F[78,x]
F[81,x] = F[13,x]*F[82,x]
F[82,x] = F[83,x]+F[92,x]
F[83,x] = F[74,x]+F[84,x]
F[84,x] = F[27,x]+F[85,x]+F[89,x]
F[85,x] = F[13,x]*F[86,x]
F[86,x] = F[13,x]+F[87,x]
F[87,x] = F[27,x]+F[85,x]+F[88,x]
F[88,x] = F[13,x]*F[74,x]
F[89,x] = F[13,x]*F[90,x]
F[90,x] = F[74,x]+F[91,x]
F[91,x] = F[89,x]
F[92,x] = F[75,x]+F[93,x]
F[93,x] = F[109,x]+F[27,x]+F[94,x]+F[98,x]
F[94,x] = F[13,x]*F[95,x]
F[95,x] = F[31,x]+F[96,x]
F[96,x] = 2*F[27,x]+F[94,x]+F[97,x]
F[97,x] = F[13,x]*F[78,x]
F[98,x] = F[13,x]*F[99,x]
F[99,x] = F[100,x]+F[103,x]
F[100,x] = F[101,x]+F[87,x]
F[101,x] = F[102,x]
F[102,x] = F[13,x]*F[91,x]
F[103,x] = F[104,x]+F[106,x]
F[104,x] = F[105,x]+F[27,x]+F[94,x]+F[98,x]
F[105,x] = F[13,x]*F[75,x]
F[106,x] = F[107,x]
F[107,x] = F[108,x]*F[13,x]
F[108,x] = F[109,x]
F[109,x] = F[110,x]*F[13,x]
F[110,x] = F[108,x]+F[75,x]
F[111,x] = F[112,x]+F[91,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]*F[13,x]
F[114,x] = F[115,x]+F[120,x]
F[115,x] = F[116,x]+F[91,x]
F[116,x] = F[117,x]
F[117,x] = F[118,x]*F[13,x]
F[118,x] = F[119,x]+F[91,x]
F[119,x] = F[117,x]
F[120,x] = F[112,x]+F[121,x]
F[121,x] = 2*F[27,x]+F[122,x]+F[123,x]+F[135,x]
F[122,x] = 0
F[123,x] = F[124,x]*F[13,x]
F[124,x] = F[125,x]+F[129,x]
F[125,x] = F[126,x]+F[127,x]
F[126,x] = F[102,x]
F[127,x] = F[128,x]
F[128,x] = F[119,x]*F[13,x]
F[129,x] = F[130,x]+F[132,x]
F[130,x] = 2*F[27,x]+F[122,x]+F[123,x]+F[131,x]
F[131,x] = F[112,x]*F[13,x]
F[132,x] = F[133,x]
F[133,x] = F[13,x]*F[134,x]
F[134,x] = F[135,x]
F[135,x] = F[13,x]*F[136,x]
F[136,x] = F[112,x]+F[134,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{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_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{5}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{72}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{39}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{23}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{23}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{47}\! \left(x \right) = F_{23}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{27}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{23}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{27}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \left(x \right)
F_{58}\! \left(x \right) = F_{13}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{64}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{49}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{13}\! \left(x \right) F_{53}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{27}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{13}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{13}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{42}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{111}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right)
F_{75}\! \left(x \right) = F_{27}\! \left(x \right)+F_{76}\! \left(x \right)+F_{81}\! \left(x \right)
F_{76}\! \left(x \right) = F_{13}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{23}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{27}\! \left(x \right)+F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{13}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{13}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{92}\! \left(x \right)
F_{83}\! \left(x \right) = F_{74}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{27}\! \left(x \right)+F_{85}\! \left(x \right)+F_{89}\! \left(x \right)
F_{85}\! \left(x \right) = F_{13}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{13}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{27}\! \left(x \right)+F_{85}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{13}\! \left(x \right) F_{74}\! \left(x \right)
F_{89}\! \left(x \right) = F_{13}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{74}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{75}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{109}\! \left(x \right)+F_{27}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)
F_{94}\! \left(x \right) = F_{13}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{31}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{94}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{13}\! \left(x \right) F_{78}\! \left(x \right)
F_{98}\! \left(x \right) = F_{13}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{103}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{87}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{13}\! \left(x \right) F_{91}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{106}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{27}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)
F_{105}\! \left(x \right) = F_{13}\! \left(x \right) F_{75}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{13}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{13}\! \left(x \right)
F_{110}\! \left(x \right) = F_{108}\! \left(x \right)+F_{75}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{91}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{13}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{120}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{91}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{13}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{91}\! \left(x \right)
F_{119}\! \left(x \right) = F_{117}\! \left(x \right)
F_{120}\! \left(x \right) = F_{112}\! \left(x \right)+F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{122}\! \left(x \right)+F_{123}\! \left(x \right)+F_{135}\! \left(x \right)
F_{122}\! \left(x \right) = 0
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{13}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{129}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{127}\! \left(x \right)
F_{126}\! \left(x \right) = F_{102}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{119}\! \left(x \right) F_{13}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{132}\! \left(x \right)
F_{130}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{122}\! \left(x \right)+F_{123}\! \left(x \right)+F_{131}\! \left(x \right)
F_{131}\! \left(x \right) = F_{112}\! \left(x \right) F_{13}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{13}\! \left(x \right) F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{13}\! \left(x \right) F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{112}\! \left(x \right)+F_{134}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_14(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_8(x))
Eq(F_8(x), F_13(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x)*F_7(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_5(x))
Eq(F_16(x), F_17(x) + F_72(x))
Eq(F_17(x), F_18(x) + F_41(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_22(x))
Eq(F_25(x), F_19(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_39(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_13(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_23(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_27(x) + F_28(x) + F_35(x))
Eq(F_35(x), F_13(x)*F_19(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_13(x)*F_40(x))
Eq(F_40(x), F_19(x) + F_38(x))
Eq(F_41(x), F_23(x) + F_42(x))
Eq(F_42(x), F_27(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_23(x))
Eq(F_45(x), F_13(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_54(x))
Eq(F_47(x), F_23(x) + F_48(x))
Eq(F_48(x), F_27(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_13(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_23(x) + F_53(x))
Eq(F_53(x), F_51(x))
Eq(F_54(x), F_42(x) + F_55(x))
Eq(F_55(x), F_27(x) + F_56(x) + F_58(x) + F_70(x))
Eq(F_56(x), F_13(x)*F_57(x))
Eq(F_57(x), F_31(x))
Eq(F_58(x), F_13(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_64(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_27(x) + F_32(x) + F_49(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_13(x)*F_53(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), F_27(x) + F_56(x) + F_58(x) + F_66(x))
Eq(F_66(x), F_13(x)*F_42(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_13(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_13(x)*F_71(x))
Eq(F_71(x), F_42(x) + F_69(x))
Eq(F_72(x), F_111(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_12(x))
Eq(F_75(x), F_27(x) + F_76(x) + F_81(x))
Eq(F_76(x), F_13(x)*F_77(x))
Eq(F_77(x), F_23(x) + F_78(x))
Eq(F_78(x), F_27(x) + F_76(x) + F_79(x))
Eq(F_79(x), F_13(x)*F_80(x))
Eq(F_80(x), F_74(x) + F_78(x))
Eq(F_81(x), F_13(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_92(x))
Eq(F_83(x), F_74(x) + F_84(x))
Eq(F_84(x), F_27(x) + F_85(x) + F_89(x))
Eq(F_85(x), F_13(x)*F_86(x))
Eq(F_86(x), F_13(x) + F_87(x))
Eq(F_87(x), F_27(x) + F_85(x) + F_88(x))
Eq(F_88(x), F_13(x)*F_74(x))
Eq(F_89(x), F_13(x)*F_90(x))
Eq(F_90(x), F_74(x) + F_91(x))
Eq(F_91(x), F_89(x))
Eq(F_92(x), F_75(x) + F_93(x))
Eq(F_93(x), F_109(x) + F_27(x) + F_94(x) + F_98(x))
Eq(F_94(x), F_13(x)*F_95(x))
Eq(F_95(x), F_31(x) + F_96(x))
Eq(F_96(x), 2*F_27(x) + F_94(x) + F_97(x))
Eq(F_97(x), F_13(x)*F_78(x))
Eq(F_98(x), F_13(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_103(x))
Eq(F_100(x), F_101(x) + F_87(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_13(x)*F_91(x))
Eq(F_103(x), F_104(x) + F_106(x))
Eq(F_104(x), F_105(x) + F_27(x) + F_94(x) + F_98(x))
Eq(F_105(x), F_13(x)*F_75(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_108(x)*F_13(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_110(x)*F_13(x))
Eq(F_110(x), F_108(x) + F_75(x))
Eq(F_111(x), F_112(x) + F_91(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x)*F_13(x))
Eq(F_114(x), F_115(x) + F_120(x))
Eq(F_115(x), F_116(x) + F_91(x))
Eq(F_116(x), F_117(x))
Eq(F_117(x), F_118(x)*F_13(x))
Eq(F_118(x), F_119(x) + F_91(x))
Eq(F_119(x), F_117(x))
Eq(F_120(x), F_112(x) + F_121(x))
Eq(F_121(x), F_122(x) + F_123(x) + F_135(x) + 2*F_27(x))
Eq(F_122(x), 0)
Eq(F_123(x), F_124(x)*F_13(x))
Eq(F_124(x), F_125(x) + F_129(x))
Eq(F_125(x), F_126(x) + F_127(x))
Eq(F_126(x), F_102(x))
Eq(F_127(x), F_128(x))
Eq(F_128(x), F_119(x)*F_13(x))
Eq(F_129(x), F_130(x) + F_132(x))
Eq(F_130(x), F_122(x) + F_123(x) + F_131(x) + 2*F_27(x))
Eq(F_131(x), F_112(x)*F_13(x))
Eq(F_132(x), F_133(x))
Eq(F_133(x), F_13(x)*F_134(x))
Eq(F_134(x), F_135(x))
Eq(F_135(x), F_13(x)*F_136(x))
Eq(F_136(x), F_112(x) + F_134(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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