0123_0132_1203_1320
Counting sequence:
1, 1, 2, 6, 20, 65, 199, 572, 1559, 4081, 10375, 25831, 63356, 153706, 369879, 884570, 2105234, 4991062, 11795846, 27806901, 65411509, 153598136, 360139845, 843356001, 1972826689, 4610785795, 10767812766, 25130195442, 58616746705, 136659657706, 318479661448, 741944382466, 1727949409400, 4023270971881, 9365522165611, 21797301326180, 50722785336795, 118016584948017, 274556104333539, 638667166102191, 1485524349894968, 3455031772413546, 8035187684735107, 18685971661619466, 43452466992474630, 101040436548244670, 234941929740030594, 546276547953305277, 1270144550602529369, 2953139144989840016, 6866038200415400105, 15963247499264284401, 37113338516742328061, 86284625301694648891, 200600579470025202714, 466366092521751852450, 1084222276323546184829, 2520616288190452773642, 5859932534032603196956, 13623111561213620962586, 31670734422112450680564, 73627209714635396081281, 171165865934280273319375, 397919100942453209359436, 925063156966975939219199, 2150537887537384595583025, 4999447954794820964244511, 11622414256625001333856247, 27019050768519410054718740, 62812103788496375232713066, 146021288167861960466516303, 339460035860685848076662586, 789152291365976612705210330, 1834563403206713330779801766, 4264856288073459071707497246, 9914615599822100079671684485, 23048740127824441253186913565, 53581930474499274325538850472, 124563096772864423857142242349, 289574509507805930787991472001, 673179945472961752204139706073, 1564955274217684885351400119091, 3638083161298031215349555670262, 8457524321097873893381292827026, 19661372795247620666458739792265, 45707174665509898491288334287850, 106256346248461834160979105718864, 247016149252272338420119967415282, 574243104010721269936291182989904, 1334953707946717016142815035448601, 3103391761412252945002387338758035, 7214512365036590268710632390108596, 16771710065597070324375556051522691, 38989502798860460164919027759410801, 90639611771884183683139077657664699, 210711562066449408180799681690190655, 489845010020281731614729813883393424, 1138751606831513390713108948664684522, 2647276540783791774185426485391631291, 6154171775235361586934310543676989546, 14306714564023477231524225950808247694
Generating function in Maple syntax:
(x^8-6*x^7+15*x^6-37*x^5+54*x^4-49*x^3+27*x^2-8*x+1)/(2*x-1)/(x^3-2*x^2+3*x-1)/(-1+x)^4
Generating function in latex syntax:
\frac{x^{8}-6 x^{7}+15 x^{6}-37 x^{5}+54 x^{4}-49 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(-1+x \right)^{4}}
Generating function in sympy syntax:
(x**8 - 6*x**7 + 15*x**6 - 37*x**5 + 54*x**4 - 49*x**3 + 27*x**2 - 8*x + 1)/((x - 1)**4*(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3-2*x^2+3*x-1)*(-1+x)^4*F(x)-x^8+6*x^7-15*x^6+37*x^5-54*x^4+49*x^3-27*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(-1+x \right)^{4} F \! \left(x \right)-x^{8}+6 x^{7}-15 x^{6}+37 x^{5}-54 x^{4}+49 x^{3}-27 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/13800*(-506*2^(1/3)*((I+6/253*23^(1/2))*3^(1/2)+18/253*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+23000-805*2^(2/3)*((I+33/161*23^(1/2))*3^(1/2)-99/161*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(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)+1/13800*(805*2^(2/3)*((I-33/161*23^(1/2))*3^(1/2)-99/161*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+23000+506*((I-6/253*23^(1/2))*3^(1/2)+18/253*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(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)+1/13800*((330*2^(2/3)*23^(1/2)*3^(1/2)-1610*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+23000+(24*23^(1/2)*3^(1/2)*2^(1/3)+1012*2^(1/3))*(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)-1/3*n^3+n^2+1/3*n-9*2^(n-1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-506 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{6 \sqrt{23}}{253}\right) \sqrt{3}+\frac{18 \,\mathrm{I} \sqrt{23}}{253}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+23000-805 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{33 \sqrt{23}}{161}\right) \sqrt{3}-\frac{99 \,\mathrm{I} \sqrt{23}}{161}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \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}}{13800}\\+\\\frac{\left(805 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{33 \sqrt{23}}{161}\right) \sqrt{3}-\frac{99 \,\mathrm{I} \sqrt{23}}{161}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+23000+506 \left(\left(\mathrm{I}-\frac{6 \sqrt{23}}{253}\right) \sqrt{3}+\frac{18 \,\mathrm{I} \sqrt{23}}{253}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \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}}{13800}\\+\\\frac{\left(\left(330 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-1610 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+23000+\left(24 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+1012 \,2^{\frac{1}{3}}\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}}{13800}\\-\frac{n^{3}}{3}+n^{2}+\frac{n}{3}-9 \,2^{n -1} & \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) = 65
a(6) = 199
a(7) = 572
a(8) = 1559
a(n+4) = -1/3*n^3+n^2+5*a(n+3)-2*a(n)+5*a(n+1)-8*a(n+2)+7/3*n+2, 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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 199
a \! \left(7\right) = 572
a \! \left(8\right) = 1559
a \! \left(n +4\right) = -\frac{n^{3}}{3}+n^{2}+5 a \! \left(n +3\right)-2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+\frac{7 n}{3}+2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14835/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[109,x]+F[13,x]+F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[36,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[29,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[18,x]
F[29,x] = F[30,x]+F[34,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[16,x]+F[33,x]
F[33,x] = F[31,x]
F[34,x] = F[35,x]
F[35,x] = F[30,x]*F[4,x]
F[36,x] = F[20,x]+F[37,x]
F[37,x] = F[105,x]+F[13,x]+F[38,x]+F[98,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[51,x]
F[40,x] = F[24,x]+F[41,x]
F[41,x] = F[13,x]+F[42,x]+F[43,x]+F[47,x]
F[42,x] = 0
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[28,x]
F[46,x] = F[34,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[30,x]+F[50,x]
F[50,x] = F[35,x]
F[51,x] = F[37,x]+F[52,x]
F[52,x] = F[13,x]+F[53,x]+F[77,x]+F[78,x]+F[95,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[59,x]
F[55,x] = F[41,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[50,x]
F[59,x] = F[52,x]+F[60,x]
F[60,x] = F[13,x]+F[61,x]+F[65,x]+F[66,x]+F[67,x]+F[76,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[56,x]
F[64,x] = F[60,x]
F[65,x] = 0
F[66,x] = 0
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[70,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[74,x]+F[75,x]
F[74,x] = F[30,x]
F[75,x] = F[71,x]
F[76,x] = 0
F[77,x] = 0
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[93,x]
F[80,x] = F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[30,x]+F[7,x]
F[87,x] = F[83,x]+F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[30,x]
F[92,x] = F[88,x]
F[93,x] = F[94,x]
F[94,x] = F[70,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[69,x]+F[71,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[100,x]+F[103,x]
F[100,x] = F[101,x]
F[101,x] = F[102,x]
F[102,x] = F[2,x]*F[4,x]
F[103,x] = F[104,x]
F[104,x] = F[82,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]+F[108,x]
F[107,x] = F[81,x]+F[83,x]
F[108,x] = F[71,x]+F[94,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[111,x]+F[112,x]
F[111,x] = F[101,x]+F[2,x]
F[112,x] = F[104,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{109}\! \left(x \right)+F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{20}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{105}\! \left(x \right)+F_{13}\! \left(x \right)+F_{38}\! \left(x \right)+F_{98}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{51}\! \left(x \right)
F_{40}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = 0
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{28}\! \left(x \right)
F_{46}\! \left(x \right) = F_{34}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{30}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{53}\! \left(x \right)+F_{77}\! \left(x \right)+F_{78}\! \left(x \right)+F_{95}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{50}\! \left(x \right)
F_{59}\! \left(x \right) = F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{76}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{56}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = 0
F_{66}\! \left(x \right) = 0
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \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_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{30}\! \left(x \right)
F_{75}\! \left(x \right) = F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = 0
F_{77}\! \left(x \right) = 0
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{93}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \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_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{87}\! \left(x \right) = F_{83}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{92}\! \left(x \right)
F_{91}\! \left(x \right) = F_{30}\! \left(x \right)
F_{92}\! \left(x \right) = F_{88}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{70}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \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_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{82}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{108}\! \left(x \right)
F_{107}\! \left(x \right) = F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{108}\! \left(x \right) = F_{71}\! \left(x \right)+F_{94}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{112}\! \left(x \right)
F_{111}\! \left(x \right) = F_{101}\! \left(x \right)+F_{2}\! \left(x \right)
F_{112}\! \left(x \right) = F_{104}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_109(x) + F_13(x) + F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_36(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_29(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_18(x))
Eq(F_29(x), F_30(x) + F_34(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_16(x) + F_33(x))
Eq(F_33(x), F_31(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_30(x)*F_4(x))
Eq(F_36(x), F_20(x) + F_37(x))
Eq(F_37(x), F_105(x) + F_13(x) + F_38(x) + F_98(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_51(x))
Eq(F_40(x), F_24(x) + F_41(x))
Eq(F_41(x), F_13(x) + F_42(x) + F_43(x) + F_47(x))
Eq(F_42(x), 0)
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_28(x))
Eq(F_46(x), F_34(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_30(x) + F_50(x))
Eq(F_50(x), F_35(x))
Eq(F_51(x), F_37(x) + F_52(x))
Eq(F_52(x), F_13(x) + F_53(x) + F_77(x) + F_78(x) + F_95(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_59(x))
Eq(F_55(x), F_41(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_50(x))
Eq(F_59(x), F_52(x) + F_60(x))
Eq(F_60(x), F_13(x) + F_61(x) + F_65(x) + F_66(x) + F_67(x) + F_76(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_56(x))
Eq(F_64(x), F_60(x))
Eq(F_65(x), 0)
Eq(F_66(x), 0)
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_70(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_74(x) + F_75(x))
Eq(F_74(x), F_30(x))
Eq(F_75(x), F_71(x))
Eq(F_76(x), 0)
Eq(F_77(x), 0)
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_93(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_30(x) + F_7(x))
Eq(F_87(x), F_83(x) + F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_92(x))
Eq(F_91(x), F_30(x))
Eq(F_92(x), F_88(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_70(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_69(x) + F_71(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_103(x))
Eq(F_100(x), F_101(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_2(x)*F_4(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_82(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x) + F_108(x))
Eq(F_107(x), F_81(x) + F_83(x))
Eq(F_108(x), F_71(x) + F_94(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_111(x) + F_112(x))
Eq(F_111(x), F_101(x) + F_2(x))
Eq(F_112(x), F_104(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 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, 3, 2, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 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, 3, 2, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 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, 3, 2, 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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [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": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [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, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 2, 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