0123_1320_2310_3021
Counting sequence:
1, 1, 2, 6, 20, 60, 139, 248, 387, 556, 755, 984, 1243, 1532, 1851, 2200, 2579, 2988, 3427, 3896, 4395, 4924, 5483, 6072, 6691, 7340, 8019, 8728, 9467, 10236, 11035, 11864, 12723, 13612, 14531, 15480, 16459, 17468, 18507, 19576, 20675, 21804, 22963, 24152, 25371, 26620, 27899, 29208, 30547, 31916, 33315, 34744, 36203, 37692, 39211, 40760, 42339, 43948, 45587, 47256, 48955, 50684, 52443, 54232, 56051, 57900, 59779, 61688, 63627, 65596, 67595, 69624, 71683, 73772, 75891, 78040, 80219, 82428, 84667, 86936, 89235, 91564, 93923, 96312, 98731, 101180, 103659, 106168, 108707, 111276, 113875, 116504, 119163, 121852, 124571, 127320, 130099, 132908, 135747, 138616, 141515
Generating function in Maple syntax:
(9*x^7-13*x^6-16*x^5-7*x^4-2*x^3-2*x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{9 x^{7}-13 x^{6}-16 x^{5}-7 x^{4}-2 x^{3}-2 x^{2}+2 x -1}{\left(x -1\right)^{3}}
Generating function in sympy syntax:
(9*x**7 - 13*x**6 - 16*x**5 - 7*x**4 - 2*x**3 - 2*x**2 + 2*x - 1)/(x - 1)**3
Implicit equation for the generating function in Maple syntax:
-(x-1)^3*F(x)+9*x^7-13*x^6-16*x^5-7*x^4-2*x^3-2*x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{3} F \! \left(x \right)+9 x^{7}-13 x^{6}-16 x^{5}-7 x^{4}-2 x^{3}-2 x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,20,15*n^2-86*n+115)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 20 & n =4 \\ 15 n^{2}-86 n +115 & \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) = 139
a(7) = 248
a(n) = 15*n^2-86*n+115, n >= 8
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) = 139
a \! \left(7\right) = 248
a \! \left(n \right) = 15 n^{2}-86 n +115, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15236/
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[48,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[14,x]+F[28,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[4,x]
F[24,x] = F[19,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[17,x]*F[4,x]
F[28,x] = F[20,x]+F[29,x]+F[45,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[35,x]
F[31,x] = F[11,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[4,x]
F[35,x] = F[28,x]+F[36,x]
F[36,x] = 2*F[20,x]+F[37,x]+F[41,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[32,x]
F[40,x] = F[36,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[23,x]*F[4,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[26,x]+F[47,x]
F[47,x] = F[44,x]
F[48,x] = F[49,x]+F[75,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[1,x]+F[26,x]
F[53,x] = F[49,x]+F[54,x]
F[54,x] = F[20,x]+F[55,x]+F[73,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[26,x]+F[58,x]
F[58,x] = F[44,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = 2*F[20,x]+F[61,x]+F[65,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[58,x]
F[64,x] = F[60,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[20,x]+F[68,x]+F[70,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[4,x]+F[43,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[52,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[67,x]+F[71,x]
F[75,x] = F[149,x]+F[20,x]+F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[91,x]
F[78,x] = F[7,x]+F[79,x]
F[79,x] = F[20,x]+F[80,x]+F[81,x]
F[80,x] = F[31,x]*F[4,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[85,x]
F[83,x] = F[26,x]+F[84,x]
F[84,x] = F[20,x]+F[45,x]+F[80,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[44,x]
F[87,x] = 2*F[20,x]+F[88,x]+F[89,x]
F[88,x] = F[39,x]*F[4,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[58,x]
F[91,x] = F[75,x]+F[92,x]
F[92,x] = F[125,x]+F[133,x]+F[20,x]+F[93,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[101,x]+F[95,x]
F[95,x] = F[79,x]+F[96,x]
F[96,x] = 2*F[20,x]+F[88,x]+F[97,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[100,x]+F[58,x]
F[100,x] = 2*F[20,x]+F[88,x]+F[89,x]
F[101,x] = F[102,x]+F[92,x]
F[102,x] = 2*F[20,x]+F[103,x]+F[107,x]+F[115,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]+F[106,x]
F[105,x] = F[96,x]
F[106,x] = F[102,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]
F[109,x] = 2*F[20,x]+F[110,x]+F[113,x]
F[110,x] = F[111,x]*F[4,x]
F[111,x] = F[112,x]+F[32,x]
F[112,x] = 2*F[20,x]+F[41,x]+F[88,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[68,x]
F[115,x] = F[116,x]*F[4,x]
F[116,x] = F[117,x]
F[117,x] = F[118,x]+F[120,x]
F[118,x] = F[119,x]
F[119,x] = F[4,x]*F[63,x]
F[120,x] = 3*F[20,x]+F[121,x]+F[123,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[100,x]
F[123,x] = F[124,x]*F[4,x]
F[124,x] = F[118,x]
F[125,x] = F[126,x]*F[4,x]
F[126,x] = F[109,x]+F[127,x]
F[127,x] = F[128,x]+F[130,x]+F[20,x]
F[128,x] = F[129,x]*F[4,x]
F[129,x] = F[11,x]+F[84,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[132,x]+F[71,x]
F[132,x] = F[68,x]
F[133,x] = F[134,x]*F[4,x]
F[134,x] = F[135,x]+F[146,x]
F[135,x] = F[136,x]+F[138,x]
F[136,x] = F[137,x]
F[137,x] = F[4,x]*F[57,x]
F[138,x] = 2*F[20,x]+F[139,x]+F[141,x]
F[139,x] = F[140,x]*F[4,x]
F[140,x] = F[100,x]+F[84,x]
F[141,x] = F[142,x]*F[4,x]
F[142,x] = F[136,x]+F[143,x]
F[143,x] = F[144,x]
F[144,x] = F[145,x]*F[4,x]
F[145,x] = F[43,x]
F[146,x] = F[147,x]+F[148,x]
F[147,x] = F[119,x]
F[148,x] = 3*F[20,x]+F[121,x]+F[123,x]
F[149,x] = F[150,x]*F[4,x]
F[150,x] = F[151,x]+F[152,x]
F[151,x] = F[127,x]+F[71,x]
F[152,x] = F[153,x]+F[154,x]
F[153,x] = F[137,x]
F[154,x] = 2*F[20,x]+F[139,x]+F[141,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_{48}\! \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_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = 0
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_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \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_{27}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)+F_{45}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \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_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{26}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{75}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{73}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{26}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{44}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \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_{58}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \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_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right)+F_{43}\! \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_{52}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{67}\! \left(x \right)+F_{71}\! \left(x \right)
F_{75}\! \left(x \right) = F_{149}\! \left(x \right)+F_{20}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{91}\! \left(x \right)
F_{78}\! \left(x \right) = F_{7}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{20}\! \left(x \right)+F_{80}\! \left(x \right)+F_{81}\! \left(x \right)
F_{80}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{85}\! \left(x \right)
F_{83}\! \left(x \right) = F_{26}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{20}\! \left(x \right)+F_{45}\! \left(x \right)+F_{80}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{44}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{58}\! \left(x \right)
F_{91}\! \left(x \right) = F_{75}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{125}\! \left(x \right)+F_{133}\! \left(x \right)+F_{20}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{101}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{79}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{88}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{58}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{92}\! \left(x \right)
F_{102}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{103}\! \left(x \right)+F_{107}\! \left(x \right)+F_{115}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{106}\! \left(x \right)
F_{105}\! \left(x \right) = F_{96}\! \left(x \right)
F_{106}\! \left(x \right) = F_{102}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{110}\! \left(x \right)+F_{113}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{4}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{32}\! \left(x \right)
F_{112}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{88}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{68}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{4}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{120}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{120}\! \left(x \right) = 3 F_{20}\! \left(x \right)+F_{121}\! \left(x \right)+F_{123}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{100}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{118}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{109}\! \left(x \right)+F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{130}\! \left(x \right)+F_{20}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right) F_{4}\! \left(x \right)
F_{129}\! \left(x \right) = F_{11}\! \left(x \right)+F_{84}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{71}\! \left(x \right)
F_{132}\! \left(x \right) = F_{68}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right) F_{4}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{146}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{138}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{138}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{139}\! \left(x \right)+F_{141}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right) F_{4}\! \left(x \right)
F_{140}\! \left(x \right) = F_{100}\! \left(x \right)+F_{84}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{4}\! \left(x \right)
F_{142}\! \left(x \right) = F_{136}\! \left(x \right)+F_{143}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right) F_{4}\! \left(x \right)
F_{145}\! \left(x \right) = F_{43}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)+F_{148}\! \left(x \right)
F_{147}\! \left(x \right) = F_{119}\! \left(x \right)
F_{148}\! \left(x \right) = 3 F_{20}\! \left(x \right)+F_{121}\! \left(x \right)+F_{123}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right) F_{4}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right)+F_{152}\! \left(x \right)
F_{151}\! \left(x \right) = F_{127}\! \left(x \right)+F_{71}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right)+F_{154}\! \left(x \right)
F_{153}\! \left(x \right) = F_{137}\! \left(x \right)
F_{154}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{139}\! \left(x \right)+F_{141}\! \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_48(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_14(x) + F_28(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_17(x)*F_4(x))
Eq(F_28(x), F_20(x) + F_29(x) + F_45(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_35(x))
Eq(F_31(x), F_11(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_4(x))
Eq(F_35(x), F_28(x) + F_36(x))
Eq(F_36(x), 2*F_20(x) + F_37(x) + F_41(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_32(x))
Eq(F_40(x), F_36(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_23(x)*F_4(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_26(x) + F_47(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_49(x) + F_75(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_1(x) + F_26(x))
Eq(F_53(x), F_49(x) + F_54(x))
Eq(F_54(x), F_20(x) + F_55(x) + F_73(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_26(x) + F_58(x))
Eq(F_58(x), F_44(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), 2*F_20(x) + F_61(x) + F_65(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_58(x))
Eq(F_64(x), F_60(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_20(x) + F_68(x) + F_70(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_4(x) + F_43(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_52(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_67(x) + F_71(x))
Eq(F_75(x), F_149(x) + F_20(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_91(x))
Eq(F_78(x), F_7(x) + F_79(x))
Eq(F_79(x), F_20(x) + F_80(x) + F_81(x))
Eq(F_80(x), F_31(x)*F_4(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_85(x))
Eq(F_83(x), F_26(x) + F_84(x))
Eq(F_84(x), F_20(x) + F_45(x) + F_80(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_44(x))
Eq(F_87(x), 2*F_20(x) + F_88(x) + F_89(x))
Eq(F_88(x), F_39(x)*F_4(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_58(x))
Eq(F_91(x), F_75(x) + F_92(x))
Eq(F_92(x), F_125(x) + F_133(x) + F_20(x) + F_93(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_101(x) + F_95(x))
Eq(F_95(x), F_79(x) + F_96(x))
Eq(F_96(x), 2*F_20(x) + F_88(x) + F_97(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_100(x) + F_58(x))
Eq(F_100(x), 2*F_20(x) + F_88(x) + F_89(x))
Eq(F_101(x), F_102(x) + F_92(x))
Eq(F_102(x), F_103(x) + F_107(x) + F_115(x) + 2*F_20(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x) + F_106(x))
Eq(F_105(x), F_96(x))
Eq(F_106(x), F_102(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_110(x) + F_113(x) + 2*F_20(x))
Eq(F_110(x), F_111(x)*F_4(x))
Eq(F_111(x), F_112(x) + F_32(x))
Eq(F_112(x), 2*F_20(x) + F_41(x) + F_88(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_68(x))
Eq(F_115(x), F_116(x)*F_4(x))
Eq(F_116(x), F_117(x))
Eq(F_117(x), F_118(x) + F_120(x))
Eq(F_118(x), F_119(x))
Eq(F_119(x), F_4(x)*F_63(x))
Eq(F_120(x), F_121(x) + F_123(x) + 3*F_20(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_100(x))
Eq(F_123(x), F_124(x)*F_4(x))
Eq(F_124(x), F_118(x))
Eq(F_125(x), F_126(x)*F_4(x))
Eq(F_126(x), F_109(x) + F_127(x))
Eq(F_127(x), F_128(x) + F_130(x) + F_20(x))
Eq(F_128(x), F_129(x)*F_4(x))
Eq(F_129(x), F_11(x) + F_84(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_71(x))
Eq(F_132(x), F_68(x))
Eq(F_133(x), F_134(x)*F_4(x))
Eq(F_134(x), F_135(x) + F_146(x))
Eq(F_135(x), F_136(x) + F_138(x))
Eq(F_136(x), F_137(x))
Eq(F_137(x), F_4(x)*F_57(x))
Eq(F_138(x), F_139(x) + F_141(x) + 2*F_20(x))
Eq(F_139(x), F_140(x)*F_4(x))
Eq(F_140(x), F_100(x) + F_84(x))
Eq(F_141(x), F_142(x)*F_4(x))
Eq(F_142(x), F_136(x) + F_143(x))
Eq(F_143(x), F_144(x))
Eq(F_144(x), F_145(x)*F_4(x))
Eq(F_145(x), F_43(x))
Eq(F_146(x), F_147(x) + F_148(x))
Eq(F_147(x), F_119(x))
Eq(F_148(x), F_121(x) + F_123(x) + 3*F_20(x))
Eq(F_149(x), F_150(x)*F_4(x))
Eq(F_150(x), F_151(x) + F_152(x))
Eq(F_151(x), F_127(x) + F_71(x))
Eq(F_152(x), F_153(x) + F_154(x))
Eq(F_153(x), F_137(x))
Eq(F_154(x), F_139(x) + F_141(x) + 2*F_20(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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 0], [0, 0]]}, {"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": [1, 3, 2, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "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, 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