0132_0321_1230_2130
Counting sequence:
1, 1, 2, 6, 20, 62, 180, 508, 1418, 3940, 10926, 30274, 83856, 232242, 643168, 1781144, 4932534, 13659656, 37827610, 104755742, 290099308, 803369830, 2224765964, 6161027412, 17061686146, 47248797036, 130845732422, 362350086426, 1003453323944, 2778855617882, 7695463616184, 21310988554672, 59016357665774, 163433548058896, 452595275065650, 1253368634803894, 3470944177404996, 9612059172476814, 26618601974829412, 73714690929416492, 204137529985896954, 565316501003103860, 1565526663952831198, 4335401020841286770, 12005992899574133056, 33248104341833943682, 92073721146768984848, 254979051998047144008, 706111539189206390950, 1955429286716341013784, 5415155372957863794506, 14996148371346456760782, 41528719027797847501724, 115005164085004949538998, 318482921598563457876156, 881972319738483271591300, 2442439201701891306436850, 6763828206965754105595516, 18731017738933854281941110, 51871664211537700046664778, 143647803102644999963648856, 397802762835331154126223018, 1101632149614886162461685096, 3050741489111972333222481248, 8448395080556361991331981790, 23396075902171999803235149344, 64790574115071609751595947618, 179424041523599191442884675430, 496877626357897964380293228148, 1375999411665166311449590956766, 3810544650161200161411364317716, 10552512165176332137164795224668, 29222991257032962561532612313194, 80927006256083755708844406032772, 224110539676394636702165401009934, 620627604030133116959184409310370, 1718699278670088469884232232954096, 4759580771656526929395677690562002, 13180670640329625435261985248042432, 36501130428002437846274510286519544, 101082301415334215032957223302078358, 279926444458329832687859144867758120, 775198162387657720876894721587715706, 2146750344119977544975782243197467518, 5944979314430604746738311152872445132, 16463385761559494416116045937007583750, 45591928314367856046585608901347773804, 126257378495974678470379091919908182964, 349643592935115673780667712795384359010, 968265328623740198918209655367592668044, 2681409771432079601444340345227301828262, 7425607578607614279195479093109697175930, 20563678293014503435060306589469382367752, 56946837071867975627429781020525751755770, 157702440501197037726424515565217570075672, 436724162724737640986904072264596340839184, 1209416884744883860861717482249097204197838, 3349228932011110979324672890047912841830192, 9274994074013186718099205260159237662132114, 25685170174773333035834660372678897348764182, 71129745382317923368729448747925367225990820
Generating function in Maple syntax:
(x^6-3*x^4+2*x^3-5*x^2+4*x-1)/(x^3-x^2+3*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{6}-3 x^{4}+2 x^{3}-5 x^{2}+4 x -1}{\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**6 - 3*x**4 + 2*x**3 - 5*x**2 + 4*x - 1)/((x - 1)**2*(x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3-x^2+3*x-1)*(x-1)^2*F(x)+x^6-3*x^4+2*x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-3 x^{4}+2 x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/7296*(((-76*I-92*19^(1/2))*3^(1/2)-276*I*19^(1/2)-76)*(1+3*19^(1/2)*3^(1/2))^(1/3)-4864+((247*I+5*19^(1/2))*3^(1/2)-15*I*19^(1/2)-247)*(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*(((-247*I+5*19^(1/2))*3^(1/2)+15*I*19^(1/2)-247)*(1+3*19^(1/2)*3^(1/2))^(2/3)-4864+((76*I-92*19^(1/2))*3^(1/2)+276*I*19^(1/2)-76)*(1+3*19^(1/2)*3^(1/2))^(1/3))*(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*((-10*19^(1/2)*3^(1/2)+494)*(1+3*19^(1/2)*3^(1/2))^(2/3)+184*(1+3*19^(1/2)*3^(1/2))^(1/3)*19^(1/2)*3^(1/2)+152*(1+3*19^(1/2)*3^(1/2))^(1/3)-4864)*(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)-n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-76 \,\mathrm{I}-92 \sqrt{19}\right) \sqrt{3}-276 \,\mathrm{I} \sqrt{19}-76\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}-4864+\left(\left(247 \,\mathrm{I}+5 \sqrt{19}\right) \sqrt{3}-15 \,\mathrm{I} \sqrt{19}-247\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(-247 \,\mathrm{I}+5 \sqrt{19}\right) \sqrt{3}+15 \,\mathrm{I} \sqrt{19}-247\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}-4864+\left(\left(76 \,\mathrm{I}-92 \sqrt{19}\right) \sqrt{3}+276 \,\mathrm{I} \sqrt{19}-76\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{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(-10 \sqrt{19}\, \sqrt{3}+494\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+184 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{19}\, \sqrt{3}+152 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}-4864\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}\\-n & \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) = 62
a(6) = 180
a(n+3) = a(n)-a(n+1)+3*a(n+2)+2*n+2, n >= 7
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) = 62
a \! \left(6\right) = 180
a \! \left(n +3\right) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 n +2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15445/
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[24,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[12,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[25,x]+F[37,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[16,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[25,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[25,x]
F[37,x] = F[18,x]+F[38,x]+F[81,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[55,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[18,x]+F[42,x]+F[47,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[10,x]+F[44,x]
F[44,x] = F[18,x]+F[22,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] = F[12,x]*F[48,x]
F[48,x] = F[46,x]+F[49,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[51,x]
F[51,x] = x^2
F[52,x] = 2*F[18,x]+F[53,x]+F[54,x]
F[53,x] = F[12,x]*F[21,x]
F[54,x] = F[12,x]*F[49,x]
F[55,x] = F[37,x]+F[56,x]
F[56,x] = 2*F[18,x]+F[57,x]+F[72,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[68,x]
F[59,x] = F[18,x]+F[60,x]+F[70,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[64,x]
F[62,x] = F[10,x]+F[63,x]
F[63,x] = F[18,x]+F[42,x]+F[45,x]
F[64,x] = F[59,x]+F[65,x]
F[65,x] = 2*F[18,x]+F[57,x]+F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[35,x]+F[68,x]
F[68,x] = 2*F[18,x]+F[66,x]+F[69,x]
F[69,x] = F[12,x]*F[59,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[25,x]+F[59,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[67,x]+F[74,x]
F[74,x] = F[75,x]+F[77,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[35,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[69,x]
F[80,x] = F[12,x]*F[74,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[71,x]+F[83,x]
F[83,x] = F[84,x]+F[94,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] = x^2
F[90,x] = F[35,x]+F[91,x]
F[91,x] = F[92,x]
F[92,x] = F[12,x]*F[35,x]
F[93,x] = F[12,x]*F[34,x]
F[94,x] = F[103,x]+F[105,x]+F[18,x]+F[95,x]
F[95,x] = F[12,x]*F[96,x]
F[96,x] = F[100,x]+F[97,x]
F[97,x] = F[44,x]+F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[12,x]*F[44,x]
F[100,x] = F[101,x]+F[68,x]
F[101,x] = F[102,x]
F[102,x] = F[12,x]*F[68,x]
F[103,x] = F[104,x]*F[12,x]
F[104,x] = F[59,x]+F[79,x]
F[105,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_{24}\! \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_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \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_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{81}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{55}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \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) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = x^{2}
F_{52}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{37}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{72}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{68}\! \left(x \right)
F_{59}\! \left(x \right) = F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{70}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{10}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{64}\! \left(x \right) = F_{59}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{35}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{25}\! \left(x \right)+F_{59}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \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_{69}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{71}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{94}\! \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) = x^{2}
F_{90}\! \left(x \right) = F_{35}\! \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_{35}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{94}\! \left(x \right) = F_{103}\! \left(x \right)+F_{105}\! \left(x \right)+F_{18}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{12}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{100}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{44}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{68}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{59}\! \left(x \right)+F_{79}\! \left(x \right)
F_{105}\! \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_24(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_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_25(x) + F_37(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_16(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_25(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_25(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_81(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_55(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_18(x) + F_42(x) + F_47(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_10(x) + F_44(x))
Eq(F_44(x), F_18(x) + F_22(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), F_12(x)*F_48(x))
Eq(F_48(x), F_46(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), x**2)
Eq(F_52(x), 2*F_18(x) + F_53(x) + F_54(x))
Eq(F_53(x), F_12(x)*F_21(x))
Eq(F_54(x), F_12(x)*F_49(x))
Eq(F_55(x), F_37(x) + F_56(x))
Eq(F_56(x), 2*F_18(x) + F_57(x) + F_72(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_68(x))
Eq(F_59(x), F_18(x) + F_60(x) + F_70(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_64(x))
Eq(F_62(x), F_10(x) + F_63(x))
Eq(F_63(x), F_18(x) + F_42(x) + F_45(x))
Eq(F_64(x), F_59(x) + F_65(x))
Eq(F_65(x), 2*F_18(x) + F_57(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_35(x) + F_68(x))
Eq(F_68(x), 2*F_18(x) + F_66(x) + F_69(x))
Eq(F_69(x), F_12(x)*F_59(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_25(x) + F_59(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_67(x) + F_74(x))
Eq(F_74(x), F_75(x) + F_77(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_35(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_69(x))
Eq(F_80(x), F_12(x)*F_74(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_71(x) + F_83(x))
Eq(F_83(x), F_84(x) + F_94(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), x**2)
Eq(F_90(x), F_35(x) + F_91(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_12(x)*F_35(x))
Eq(F_93(x), F_12(x)*F_34(x))
Eq(F_94(x), F_103(x) + F_105(x) + F_18(x) + F_95(x))
Eq(F_95(x), F_12(x)*F_96(x))
Eq(F_96(x), F_100(x) + F_97(x))
Eq(F_97(x), F_44(x) + F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_12(x)*F_44(x))
Eq(F_100(x), F_101(x) + F_68(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_12(x)*F_68(x))
Eq(F_103(x), F_104(x)*F_12(x))
Eq(F_104(x), F_59(x) + F_79(x))
Eq(F_105(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 1, 2], "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": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 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, 1, 2], "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": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [2, 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": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 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": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "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": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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