0132_0321_1023_1230
Counting sequence:
1, 1, 2, 6, 20, 60, 167, 456, 1229, 3282, 8712, 23029, 60697, 159655, 419359, 1100429, 2885620, 7563222, 19816537, 51909364, 135953813, 356029800, 932278546, 2441068781, 6391411425, 16734055025, 43812389753, 114706123497, 300311515636, 786238603676, 2058423019819, 5389064895372, 14108835010581, 36937556644674, 96704049215620, 253174985146953, 662821631170489, 1735291241746711, 4543054546541859, 11893876908678529, 31138584476147804, 81521891779690838, 213427118930304405, 558759516635182104, 1462851525926507285, 3829795235786944552, 10026534502652156278, 26249808862981224369, 68722893172963651625, 179918872654611395297, 471033728467056033941, 1233182319508116005789, 3228513242493738447156, 8452357430847290598348, 22128559092120330373551, 57933319922896535274368, 151671400818898498489949, 397080882795583215013602, 1039571248049347459161440, 2721632862238068952939821, 7125327340293749757555833, 18654349161639176780705175, 48837720150134277193904103, 127858811298899038229226853, 334738713765204713992318084, 876357330031002860283833158, 2294333276390868883322048497, 6006642499257598439179826828, 15725594221595273856713919349, 41170140165920630219418800184, 107784826276888365961993352026, 282184338666071971337965483045, 738768189723769207972215064049, 1934120230509726565330846774017, 5063592501813670564364208517009, 13256657274946477776778141067873, 34706379323053706404082627301396, 90862480694266037798942399000972, 237881062759838939643346002771827, 622780707585424653783282113164260, 1630461059996754823372760931805669, 4268602472405428023304049214257522, 11175346357220611127826882341906140, 29257436599258395250099402569489633, 76596963440558234600650674287535513, 200533453722423040301242269603057639, 525003397726723267920567937510579627, 1374476739457769536805522344148536953, 3598426820646627229207941348453769900, 9420803722482189192492796558940309350, 24663984346800082050001946240833289133, 64571149317918317587630977187271964304, 169049463606955350086414913114888677621, 442577241502948614376987122887478679640, 1158682260901892114753561679095250452478, 3033469541202730712671610426469966453897, 7941726362706305509463570696664130387577, 20791709546916195906418330495491303054481, 54433402278042300769480863230199832375981, 142508497287210740538615231563816607497589, 373092089583589983633344475102317335486676
Generating function in Maple syntax:
(x^9+2*x^8+2*x^7-5*x^6-4*x^5+3*x^4+x^3+4*x^2-4*x+1)/(x-1)/(x^2-3*x+1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{x^{9}+2 x^{8}+2 x^{7}-5 x^{6}-4 x^{5}+3 x^{4}+x^{3}+4 x^{2}-4 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x**9 + 2*x**8 + 2*x**7 - 5*x**6 - 4*x**5 + 3*x**4 + x**3 + 4*x**2 - 4*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2-3*x+1)*(x^3+x^2+x-1)*F(x)+x^9+2*x^8+2*x^7-5*x^6-4*x^5+3*x^4+x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{9}+2 x^{8}+2 x^{7}-5 x^{6}-4 x^{5}+3 x^{4}+x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,n = 3,6,1/660*(((5*11^(1/2)+25*I)*3^(1/2)+15*I*11^(1/2)+25)*(17+3*11^(1/2)*3^(1/2))^(1/3)-70+((-40*11^(1/2)-230*I)*3^(1/2)+120*I*11^(1/2)+230)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/660*(((-40*11^(1/2)+230*I)*3^(1/2)-120*I*11^(1/2)+230)*(17+3*11^(1/2)*3^(1/2))^(2/3)-70+((5*11^(1/2)-25*I)*3^(1/2)-15*I*11^(1/2)+25)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/660*((80*11^(1/2)*3^(1/2)-460)*(17+3*11^(1/2)*3^(1/2))^(2/3)-10*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-50*(17+3*11^(1/2)*3^(1/2))^(1/3)-70)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/660*(5958*5^(1/2)-12930)*(3/2-1/2*5^(1/2))^(-n)+1/2+1/660*(-5958*5^(1/2)-12930)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(\left(5 \sqrt{11}+25 \,\mathrm{I}\right) \sqrt{3}+15 \,\mathrm{I} \sqrt{11}+25\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-70+\left(\left(-40 \sqrt{11}-230 \,\mathrm{I}\right) \sqrt{3}+120 \,\mathrm{I} \sqrt{11}+230\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{660}\\+\\\frac{\left(\left(\left(-40 \sqrt{11}+230 \,\mathrm{I}\right) \sqrt{3}-120 \,\mathrm{I} \sqrt{11}+230\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-70+\left(\left(5 \sqrt{11}-25 \,\mathrm{I}\right) \sqrt{3}-15 \,\mathrm{I} \sqrt{11}+25\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{660}\\+\\\frac{\left(\left(80 \sqrt{11}\, \sqrt{3}-460\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-10 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-50 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-70\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{660}\\+\frac{\left(5958 \sqrt{5}-12930\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{660}+\frac{1}{2}+\\\frac{\left(-5958 \sqrt{5}-12930\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{660} & \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) = 60
a(6) = 167
a(7) = 456
a(8) = 1229
a(9) = 3282
a(n+5) = a(n)-2*a(n+1)-a(n+2)-3*a(n+3)+4*a(n+4)+1, n >= 10
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) = 60
a \! \left(6\right) = 167
a \! \left(7\right) = 456
a \! \left(8\right) = 1229
a \! \left(9\right) = 3282
a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right)+1, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15414/
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[22,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[20,x]*F[4,x]
F[20,x] = F[11,x]+F[21,x]
F[21,x] = F[16,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[114,x]+F[15,x]+F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[41,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[1,x]+F[4,x]
F[30,x] = F[15,x]+F[31,x]+F[39,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[17,x]+F[33,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[15,x]+F[16,x]+F[35,x]
F[35,x] = F[11,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[18,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[11,x]+F[18,x]
F[41,x] = F[42,x]+F[95,x]
F[42,x] = F[15,x]+F[24,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[2,x]+F[45,x]
F[45,x] = F[15,x]+F[46,x]+F[94,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[4,x]+F[49,x]
F[49,x] = F[15,x]+F[31,x]+F[35,x]
F[50,x] = F[45,x]+F[51,x]
F[51,x] = F[15,x]+F[52,x]+F[81,x]+F[89,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[59,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[56,x]
F[56,x] = x^2
F[57,x] = F[58,x]
F[58,x] = F[18,x]*F[4,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = 2*F[15,x]+F[52,x]+F[61,x]
F[61,x] = F[4,x]*F[45,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,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[76,x]
F[70,x] = F[4,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[17,x]+F[74,x]
F[74,x] = F[18,x]+F[75,x]
F[75,x] = F[37,x]
F[76,x] = F[67,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[66,x]+F[80,x]
F[80,x] = F[65,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[66,x]+F[83,x]
F[83,x] = F[84,x]
F[84,x] = 2*F[15,x]+F[65,x]+F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[56,x]
F[88,x] = F[85,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[15,x]+F[46,x]+F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[2,x]+F[93,x]
F[93,x] = F[68,x]
F[94,x] = F[2,x]*F[4,x]
F[95,x] = F[112,x]+F[15,x]+F[81,x]+F[96,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[105,x]+F[98,x]
F[98,x] = F[102,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[4,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[18,x]
F[105,x] = F[106,x]+F[109,x]
F[106,x] = 2*F[15,x]+F[107,x]+F[96,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[45,x]
F[109,x] = F[110,x]
F[110,x] = F[111,x]*F[4,x]
F[111,x] = F[64,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[64,x]+F[90,x]
F[114,x] = F[115,x]*F[4,x]
F[115,x] = F[116,x]+F[92,x]
F[116,x] = F[117,x]+F[90,x]
F[117,x] = 2*F[15,x]+F[65,x]+F[85,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_{22}\! \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_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{114}\! \left(x \right)+F_{15}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{41}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{39}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{95}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{94}\! \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_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{50}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{81}\! \left(x \right)+F_{89}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = x^{2}
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \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_{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_{76}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{18}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{37}\! \left(x \right)
F_{76}\! \left(x \right) = F_{67}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{66}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{65}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{66}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{56}\! \left(x \right)
F_{88}\! \left(x \right) = F_{85}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{2}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{68}\! \left(x \right)
F_{94}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{95}\! \left(x \right) = F_{112}\! \left(x \right)+F_{15}\! \left(x \right)+F_{81}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{105}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{102}\! \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_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{18}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{109}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{107}\! \left(x \right)+F_{96}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{45}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{4}\! \left(x \right)
F_{111}\! \left(x \right) = F_{64}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{64}\! \left(x \right)+F_{90}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{4}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{92}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{90}\! \left(x \right)
F_{117}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{85}\! \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_22(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_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_21(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_114(x) + F_15(x) + F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_41(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_1(x) + F_4(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_39(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_17(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_15(x) + F_16(x) + F_35(x))
Eq(F_35(x), F_11(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_18(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_11(x) + F_18(x))
Eq(F_41(x), F_42(x) + F_95(x))
Eq(F_42(x), F_15(x) + F_24(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_2(x) + F_45(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_94(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_4(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_31(x) + F_35(x))
Eq(F_50(x), F_45(x) + F_51(x))
Eq(F_51(x), F_15(x) + F_52(x) + F_81(x) + F_89(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_59(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), x**2)
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_18(x)*F_4(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), 2*F_15(x) + F_52(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_45(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(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_76(x))
Eq(F_70(x), F_4(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_17(x) + F_74(x))
Eq(F_74(x), F_18(x) + F_75(x))
Eq(F_75(x), F_37(x))
Eq(F_76(x), F_67(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_66(x) + F_80(x))
Eq(F_80(x), F_65(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_66(x) + F_83(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), 2*F_15(x) + F_65(x) + F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_56(x))
Eq(F_88(x), F_85(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_15(x) + F_46(x) + F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_2(x) + F_93(x))
Eq(F_93(x), F_68(x))
Eq(F_94(x), F_2(x)*F_4(x))
Eq(F_95(x), F_112(x) + F_15(x) + F_81(x) + F_96(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_105(x) + F_98(x))
Eq(F_98(x), F_102(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_4(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_18(x))
Eq(F_105(x), F_106(x) + F_109(x))
Eq(F_106(x), F_107(x) + 2*F_15(x) + F_96(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_45(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_111(x)*F_4(x))
Eq(F_111(x), F_64(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_64(x) + F_90(x))
Eq(F_114(x), F_115(x)*F_4(x))
Eq(F_115(x), F_116(x) + F_92(x))
Eq(F_116(x), F_117(x) + F_90(x))
Eq(F_117(x), 2*F_15(x) + F_65(x) + F_85(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, 2, 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, 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, 2, 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, 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, 2, 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, 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, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"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, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"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, 0], [1, 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