0132_0321_1023_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 52, 138, 362, 939, 2410, 6130, 15476, 38826, 96882, 240619, 595162, 1466787, 3603290, 8826289, 21563956, 52560198, 127836892, 310317645, 751931352, 1819002770, 4393663990, 10597597778, 25528113746, 61418587285, 147599853732, 354330755541, 849762246382, 2036000592863, 4873872629860, 11657572793522, 27861201577646, 66537692269487, 158792066043302, 378702980834882, 902593120986952, 2149919210496666, 5118015328905250, 12177046694347071, 28957038484694206, 68825315884473079, 163505979158620002, 388256982400942493, 921538374327759012, 2186371131374680654, 5185110109067409760, 12292001963862818225, 29128914411713671476, 69003187668172415586, 163404111211333085482, 386822079012015287074, 915417409308422577762, 2165666764654267643177, 5121926097231971503240, 12110124840261272802409, 28624754594464256562006, 67642049916315592578283, 159800226210504972359892, 377422061895087612855450, 891189156781549886636402, 2103819697374669793398835, 4965309195584845204248290, 11716185538554890492521298, 27639565905526079020181596, 65190449761615144577490570, 153725937014488984342356882, 362429259562717938927687571, 854308446753936787764374882, 2013371004878531281132606603, 4744088718344105246712604010, 11176448203868001257768047817, 26325636586731092168925571796, 61998329405587726517349011222, 145985264049493314386454314852, 343690716068372609086184079637, 809017813674867716831188411152, 1904064073940447886521541966706, 4480637931367310197566306894750, 10542279584534846136421061009266, 24800880898075634615112361849906, 58336321627869770076151824403389, 137199251444137038223937007009452, 322631351881754434504754466230973, 758586016732937619380465566553662, 1783396073881806594418613578808119, 4192149043279245661082316500553860, 9853096698804248948150491951893570, 23155653653294096922683852499852726, 54411503417450714564130122644156471, 127842384641142344811468364436503582, 300338147677304823021098260147397154, 705502634792285240380003666536520112, 1657067757664606563467585411019960442, 3891672874775284091379993436920298690, 9138769844477883022992162343401474279, 21458304160238321382032143145512292806, 50380232044917196776437718821537851743
Generating function in Maple syntax:
(x^8-3*x^7+6*x^6-6*x^5+5*x^4-7*x^3+9*x^2-5*x+1)/(x^3-2*x^2+3*x-1)^2
Generating function in latex syntax:
\frac{x^{8}-3 x^{7}+6 x^{6}-6 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}
Generating function in sympy syntax:
(x**8 - 3*x**7 + 6*x**6 - 6*x**5 + 5*x**4 - 7*x**3 + 9*x**2 - 5*x + 1)/(x**3 - 2*x**2 + 3*x - 1)**2
Implicit equation for the generating function in Maple syntax:
-(x^3-2*x^2+3*x-1)^2*F(x)+x^8-3*x^7+6*x^6-6*x^5+5*x^4-7*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+x^{8}-3 x^{7}+6 x^{6}-6 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 1,1,n = 2,2,1/158700*(46*2^(1/3)*(((2*n+227/23)*23^(1/2)+I*n-13*I)*3^(1/2)+(6*I*n+681/23*I)*23^(1/2)+n-13)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1955*(((1/17*n+64/391)*23^(1/2)+I*n+76/17*I)*3^(1/2)+(-3/17*I*n-192/391*I)*23^(1/2)-n-76/17)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+29900*n+52900)*(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)+1/158700*(-1955*(((-1/17*n-64/391)*23^(1/2)+I*n+76/17*I)*3^(1/2)+(-3/17*I*n-192/391*I)*23^(1/2)+n+76/17)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)-46*(((-2*n-227/23)*23^(1/2)+I*n-13*I)*3^(1/2)+(6*I*n+681/23*I)*23^(1/2)-n+13)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+29900*n+52900)*(-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)-2/1725*(5/4*2^(2/3)*(3^(1/2)*(n+64/23)*23^(1/2)-17*n-76)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*(3^(1/2)*(n+227/46)*23^(1/2)+1/2*n-13/2)*(11+3*23^(1/2)*3^(1/2))^(2/3)-325/2*n-575/2)*(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))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =1 \\ 2 & n =2 \\ \frac{\left(46 \,2^{\frac{1}{3}} \left(\left(\left(2 n +\frac{227}{23}\right) \sqrt{23}+\mathrm{I} n -13 \,\mathrm{I}\right) \sqrt{3}+\left(6 \,\mathrm{I} n +\frac{681 \,\mathrm{I}}{23}\right) \sqrt{23}+n -13\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+1955 \left(\left(\left(\frac{n}{17}+\frac{64}{391}\right) \sqrt{23}+\mathrm{I} n +\frac{76 \,\mathrm{I}}{17}\right) \sqrt{3}+\left(-\frac{3 \,\mathrm{I} n}{17}-\frac{192 \,\mathrm{I}}{391}\right) \sqrt{23}-n -\frac{76}{17}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+29900 n +52900\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}}{158700}\\+\\\frac{\left(-1955 \left(\left(\left(-\frac{n}{17}-\frac{64}{391}\right) \sqrt{23}+\mathrm{I} n +\frac{76 \,\mathrm{I}}{17}\right) \sqrt{3}+\left(-\frac{3 \,\mathrm{I} n}{17}-\frac{192 \,\mathrm{I}}{391}\right) \sqrt{23}+n +\frac{76}{17}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-46 \left(\left(\left(-2 n -\frac{227}{23}\right) \sqrt{23}+\mathrm{I} n -13 \,\mathrm{I}\right) \sqrt{3}+\left(6 \,\mathrm{I} n +\frac{681 \,\mathrm{I}}{23}\right) \sqrt{23}-n +13\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+29900 n +52900\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}}{158700}\\-\\\frac{2 \left(\frac{5 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n +\frac{64}{23}\right) \sqrt{23}-17 n -76\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{4}+2^{\frac{1}{3}} \left(\sqrt{3}\, \left(n +\frac{227}{46}\right) \sqrt{23}+\frac{n}{2}-\frac{13}{2}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{325 n}{2}-\frac{575}{2}\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}}{1725} & \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) = 52
a(6) = 138
a(7) = 362
a(8) = 939
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5), n >= 9
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) = 52
a \! \left(6\right) = 138
a \! \left(7\right) = 362
a \! \left(8\right) = 939
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19326/
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[20,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[21,x]+F[41,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[21,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[36,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[15,x]+F[42,x]+F[84,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[58,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[1,x]+F[4,x]
F[48,x] = F[15,x]+F[49,x]+F[56,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[17,x]+F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[15,x]+F[16,x]+F[53,x]
F[53,x] = F[11,x]*F[4,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[51,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[11,x]
F[58,x] = F[59,x]+F[64,x]
F[59,x] = F[15,x]+F[42,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[21,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[21,x]*F[4,x]
F[64,x] = 2*F[15,x]+F[65,x]+F[82,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[78,x]
F[67,x] = F[68,x]+F[76,x]
F[68,x] = F[15,x]+F[63,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[73,x]
F[71,x] = F[4,x]+F[72,x]
F[72,x] = F[15,x]+F[49,x]+F[53,x]
F[73,x] = F[68,x]+F[74,x]
F[74,x] = 2*F[15,x]+F[65,x]+F[75,x]
F[75,x] = F[36,x]*F[4,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[67,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = 2*F[15,x]+F[75,x]+F[77,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[78,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[36,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[89,x]
F[86,x] = F[21,x]+F[87,x]
F[87,x] = F[15,x]+F[69,x]+F[88,x]
F[88,x] = F[4,x]*F[86,x]
F[89,x] = F[36,x]+F[90,x]
F[90,x] = 2*F[15,x]+F[77,x]+F[91,x]
F[91,x] = F[4,x]*F[89,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_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{84}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{58}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{21}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{82}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{78}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{76}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{73}\! \left(x \right) = F_{68}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{36}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{89}\! \left(x \right)
F_{86}\! \left(x \right) = F_{21}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{15}\! \left(x \right)+F_{69}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{89}\! \left(x \right) = F_{36}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{77}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \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_20(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_41(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_31(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_21(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_36(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_84(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_58(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_1(x) + F_4(x))
Eq(F_48(x), F_15(x) + F_49(x) + F_56(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_17(x) + F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_15(x) + F_16(x) + F_53(x))
Eq(F_53(x), F_11(x)*F_4(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_51(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_11(x))
Eq(F_58(x), F_59(x) + F_64(x))
Eq(F_59(x), F_15(x) + F_42(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_21(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_21(x)*F_4(x))
Eq(F_64(x), 2*F_15(x) + F_65(x) + F_82(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_78(x))
Eq(F_67(x), F_68(x) + F_76(x))
Eq(F_68(x), F_15(x) + F_63(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_73(x))
Eq(F_71(x), F_4(x) + F_72(x))
Eq(F_72(x), F_15(x) + F_49(x) + F_53(x))
Eq(F_73(x), F_68(x) + F_74(x))
Eq(F_74(x), 2*F_15(x) + F_65(x) + F_75(x))
Eq(F_75(x), F_36(x)*F_4(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_67(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), 2*F_15(x) + F_75(x) + F_77(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_78(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_36(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_89(x))
Eq(F_86(x), F_21(x) + F_87(x))
Eq(F_87(x), F_15(x) + F_69(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_86(x))
Eq(F_89(x), F_36(x) + F_90(x))
Eq(F_90(x), 2*F_15(x) + F_77(x) + F_91(x))
Eq(F_91(x), F_4(x)*F_89(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"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": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "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": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], 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