0132_0231_0321_2310
Counting sequence:
1, 1, 2, 6, 20, 65, 195, 540, 1402, 3467, 8269, 19198, 43676, 97829, 216471, 474352, 1031214, 2227023, 4782673, 10221874, 21755888, 46134921, 97514747, 205517636, 432009570, 905965395, 1895820565, 3959417510, 8254384132, 17179862317, 35701907999, 74088177368, 153545071446, 317827569559, 657129984921, 1357209653082, 2800318663384, 5772436030993, 11888469459203, 24464133700524, 50302656951818, 103354092990491, 212205744138717, 435406604575566, 892803441728620, 1829587348591925, 3747135627431335, 7670193115334080, 15692229951585662, 32088147344979167, 65583669573544929, 133982088914231938, 273593677362714816, 558446353793896089, 1139410705724687371, 2323857407723125012, 4737786807993707954, 9655717601082286563, 19671723172354266533, 40064022285087869174, 81569196450934356948, 166020696663385894461, 337806000849806090287, 687141216745680720296, 1397340863583498453670, 2840798587351270863655, 5773830895071089566505, 11732129230879274734250, 23833193343232740589992, 48404256449413863338017, 98284252424724490903059, 199519983901242510166908, 404942925906072076957786, 821691768019318267061419, 1666995368452984760307821, 3381214401734665972874142, 6856876133126724850148924, 13902646925568235508977733, 28183083169766042635188663, 57121744976791228504711824, 115754647228100743477955278, 234531609005238059892830831, 475107847108549265659353457, 962304952413244823065935826, 1948788421218782229626168720, 3945933875222149626240764585, 7988581816013469586458210075, 16170591763165279840869602020, 32728039788607241017645381122, 66229792101767844707102922867, 134007009252642414757829966389, 271108868603498280202907966278, 548407437403423461780311784356, 1109194275199700726309615049549, 2243147351185109058117212830271, 4535812303941633327230390884472, 9170659811026097076452711970294, 18539390028337854996889284088503, 37474920869247031681746288209593, 75742123363636706739428016212474, 153068809977558700230726911730808
Generating function in Maple syntax:
-(3*x^6+6*x^5-21*x^4+28*x^3-20*x^2+7*x-1)/(2*x-1)^2/(-1+x)^4
Generating function in latex syntax:
-\frac{3 x^{6}+6 x^{5}-21 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{4}}
Generating function in sympy syntax:
(-3*x**6 - 6*x**5 + 21*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(-1+x)^4*F(x)+3*x^6+6*x^5-21*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(-1+x \right)^{4} F \! \left(x \right)+3 x^{6}+6 x^{5}-21 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,6-1/6*n+3/2*n^2-1/3*n^3+5/4*2^n*n-17/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 6-\frac{n}{6}+\frac{3 n^{2}}{2}-\frac{n^{3}}{3}+\frac{5 \,2^{n} n}{4}-\frac{17 \,2^{n}}{4} & \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) = 195
a(n+2) = -1/3*n^3+7/2*n^2-4*a(n)+4*a(n+1)-37/6*n+5, 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) = 65
a \! \left(6\right) = 195
a \! \left(n +2\right) = -\frac{n^{3}}{3}+\frac{7 n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{37 n}{6}+5, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15316/
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[18,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[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[42,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[5,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[15,x]+F[27,x]+F[29,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[26,x]+F[6,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[36,x]
F[31,x] = F[23,x]+F[32,x]
F[32,x] = F[15,x]+F[33,x]+F[35,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[10,x]+F[32,x]
F[35,x] = F[12,x]*F[31,x]
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[23,x]
F[39,x] = 2*F[15,x]+F[40,x]+F[41,x]
F[40,x] = F[12,x]*F[32,x]
F[41,x] = F[12,x]*F[36,x]
F[42,x] = F[15,x]+F[43,x]+F[88,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[58,x]
F[45,x] = F[46,x]+F[6,x]
F[46,x] = F[15,x]+F[47,x]+F[56,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[10,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[49,x]
F[52,x] = F[14,x]+F[53,x]
F[53,x] = 2*F[15,x]+F[54,x]+F[55,x]
F[54,x] = F[12,x]*F[50,x]
F[55,x] = F[12,x]*F[52,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[13,x]
F[58,x] = F[59,x]+F[72,x]
F[59,x] = F[15,x]+F[60,x]+F[62,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[59,x]+F[6,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[31,x]+F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[15,x]+F[38,x]+F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[12,x]+F[65,x]
F[68,x] = F[15,x]+F[40,x]+F[69,x]+F[71,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[14,x]+F[68,x]
F[71,x] = F[12,x]*F[64,x]
F[72,x] = F[15,x]+F[73,x]+F[75,x]+F[86,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[46,x]+F[72,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[77,x]+F[82,x]
F[77,x] = F[32,x]+F[78,x]
F[78,x] = 2*F[15,x]+F[79,x]+F[81,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[50,x]+F[78,x]
F[81,x] = F[12,x]*F[77,x]
F[82,x] = F[39,x]+F[83,x]
F[83,x] = 3*F[15,x]+F[84,x]+F[85,x]
F[84,x] = F[12,x]*F[78,x]
F[85,x] = F[12,x]*F[82,x]
F[86,x] = F[12,x]*F[87,x]
F[87,x] = F[64,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[101,x]+F[90,x]
F[90,x] = F[19,x]+F[91,x]
F[91,x] = F[100,x]+F[15,x]+F[92,x]
F[92,x] = F[12,x]*F[93,x]
F[93,x] = F[94,x]+F[96,x]
F[94,x] = F[10,x]+F[95,x]
F[95,x] = F[15,x]+F[17,x]+F[47,x]
F[96,x] = F[32,x]+F[97,x]
F[97,x] = F[15,x]+F[71,x]+F[75,x]+F[98,x]
F[98,x] = F[12,x]*F[99,x]
F[99,x] = F[95,x]+F[97,x]
F[100,x] = F[12,x]*F[90,x]
F[101,x] = F[102,x]+F[137,x]
F[102,x] = F[103,x]+F[15,x]+F[38,x]
F[103,x] = F[104,x]*F[12,x]
F[104,x] = F[105,x]+F[119,x]
F[105,x] = F[106,x]+F[12,x]
F[106,x] = F[107,x]+F[115,x]+F[15,x]
F[107,x] = F[108,x]*F[12,x]
F[108,x] = F[109,x]+F[112,x]
F[109,x] = F[110,x]+F[12,x]
F[110,x] = F[111,x]
F[111,x] = F[109,x]*F[12,x]
F[112,x] = F[113,x]+F[116,x]
F[113,x] = F[114,x]+F[115,x]+F[15,x]
F[114,x] = x^2
F[115,x] = x^2
F[116,x] = 2*F[15,x]+F[117,x]+F[118,x]
F[117,x] = F[110,x]*F[12,x]
F[118,x] = F[112,x]*F[12,x]
F[119,x] = F[120,x]+F[65,x]
F[120,x] = F[121,x]+F[123,x]+F[133,x]+F[15,x]
F[121,x] = F[12,x]*F[122,x]
F[122,x] = F[106,x]+F[120,x]
F[123,x] = F[12,x]*F[124,x]
F[124,x] = F[125,x]+F[130,x]
F[125,x] = F[126,x]+F[65,x]
F[126,x] = 2*F[15,x]+F[127,x]+F[129,x]
F[127,x] = F[12,x]*F[128,x]
F[128,x] = F[110,x]+F[126,x]
F[129,x] = F[12,x]*F[125,x]
F[130,x] = F[131,x]+F[134,x]
F[131,x] = 2*F[15,x]+F[132,x]+F[133,x]
F[132,x] = F[12,x]*F[65,x]
F[133,x] = F[12,x]*F[37,x]
F[134,x] = 3*F[15,x]+F[135,x]+F[136,x]
F[135,x] = F[12,x]*F[126,x]
F[136,x] = F[12,x]*F[130,x]
F[137,x] = F[138,x]+F[15,x]+F[178,x]+F[40,x]
F[138,x] = F[12,x]*F[139,x]
F[139,x] = F[140,x]+F[156,x]
F[140,x] = F[14,x]+F[141,x]
F[141,x] = F[142,x]+F[15,x]+F[151,x]+F[155,x]
F[142,x] = F[12,x]*F[143,x]
F[143,x] = F[144,x]+F[148,x]
F[144,x] = F[145,x]+F[146,x]
F[145,x] = F[16,x]
F[146,x] = F[147,x]
F[147,x] = F[12,x]*F[144,x]
F[148,x] = F[149,x]+F[152,x]
F[149,x] = 2*F[15,x]+F[150,x]+F[151,x]
F[150,x] = F[12,x]*F[145,x]
F[151,x] = F[12,x]*F[14,x]
F[152,x] = 3*F[15,x]+F[153,x]+F[154,x]
F[153,x] = F[12,x]*F[146,x]
F[154,x] = F[12,x]*F[148,x]
F[155,x] = 0
F[156,x] = F[157,x]+F[68,x]
F[157,x] = F[15,x]+F[158,x]+F[160,x]+F[173,x]+F[177,x]
F[158,x] = F[12,x]*F[159,x]
F[159,x] = F[141,x]+F[157,x]
F[160,x] = F[12,x]*F[161,x]
F[161,x] = F[162,x]+F[170,x]
F[162,x] = F[163,x]+F[166,x]
F[163,x] = 2*F[15,x]+F[164,x]+F[40,x]
F[164,x] = F[12,x]*F[165,x]
F[165,x] = F[145,x]+F[163,x]
F[166,x] = 3*F[15,x]+F[167,x]+F[169,x]
F[167,x] = F[12,x]*F[168,x]
F[168,x] = F[146,x]+F[166,x]
F[169,x] = F[12,x]*F[162,x]
F[170,x] = F[171,x]+F[174,x]
F[171,x] = 3*F[15,x]+F[172,x]+F[173,x]
F[172,x] = F[12,x]*F[163,x]
F[173,x] = F[12,x]*F[39,x]
F[174,x] = 4*F[15,x]+F[175,x]+F[176,x]
F[175,x] = F[12,x]*F[166,x]
F[176,x] = F[12,x]*F[170,x]
F[177,x] = 0
F[178,x] = F[101,x]*F[12,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_{18}\! \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_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{5}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{88}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{58}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{72}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{59}\! \left(x \right)+F_{6}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{31}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \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_{12}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{14}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{72}\! \left(x \right) = F_{15}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)+F_{86}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{46}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{82}\! \left(x \right)
F_{77}\! \left(x \right) = F_{32}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{50}\! \left(x \right)+F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{82}\! \left(x \right) = F_{39}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{86}\! \left(x \right) = F_{12}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{64}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{101}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{19}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{100}\! \left(x \right)+F_{15}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{94}\! \left(x \right) = F_{10}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{47}\! \left(x \right)
F_{96}\! \left(x \right) = F_{32}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{15}\! \left(x \right)+F_{71}\! \left(x \right)+F_{75}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{12}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{95}\! \left(x \right)+F_{97}\! \left(x \right)
F_{100}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{137}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{15}\! \left(x \right)+F_{38}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{119}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{115}\! \left(x \right)+F_{15}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{12}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{112}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{12}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{116}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{115}\! \left(x \right)+F_{15}\! \left(x \right)
F_{114}\! \left(x \right) = x^{2}
F_{115}\! \left(x \right) = x^{2}
F_{116}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{117}\! \left(x \right)+F_{118}\! \left(x \right)
F_{117}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{118}\! \left(x \right) = F_{112}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{65}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{123}\! \left(x \right)+F_{133}\! \left(x \right)+F_{15}\! \left(x \right)
F_{121}\! \left(x \right) = F_{12}\! \left(x \right) F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{106}\! \left(x \right)+F_{120}\! \left(x \right)
F_{123}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{130}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{65}\! \left(x \right)
F_{126}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{127}\! \left(x \right)+F_{129}\! \left(x \right)
F_{127}\! \left(x \right) = F_{12}\! \left(x \right) F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{110}\! \left(x \right)+F_{126}\! \left(x \right)
F_{129}\! \left(x \right) = F_{12}\! \left(x \right) F_{125}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{134}\! \left(x \right)
F_{131}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{132}\! \left(x \right)+F_{133}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{133}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{134}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{135}\! \left(x \right)+F_{136}\! \left(x \right)
F_{135}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{136}\! \left(x \right) = F_{12}\! \left(x \right) F_{130}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)+F_{15}\! \left(x \right)+F_{178}\! \left(x \right)+F_{40}\! \left(x \right)
F_{138}\! \left(x \right) = F_{12}\! \left(x \right) F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right)+F_{156}\! \left(x \right)
F_{140}\! \left(x \right) = F_{14}\! \left(x \right)+F_{141}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{15}\! \left(x \right)+F_{151}\! \left(x \right)+F_{155}\! \left(x \right)
F_{142}\! \left(x \right) = F_{12}\! \left(x \right) F_{143}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)+F_{148}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right)+F_{146}\! \left(x \right)
F_{145}\! \left(x \right) = F_{16}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)
F_{147}\! \left(x \right) = F_{12}\! \left(x \right) F_{144}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{152}\! \left(x \right)
F_{149}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{150}\! \left(x \right)+F_{151}\! \left(x \right)
F_{150}\! \left(x \right) = F_{12}\! \left(x \right) F_{145}\! \left(x \right)
F_{151}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{152}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{153}\! \left(x \right)+F_{154}\! \left(x \right)
F_{153}\! \left(x \right) = F_{12}\! \left(x \right) F_{146}\! \left(x \right)
F_{154}\! \left(x \right) = F_{12}\! \left(x \right) F_{148}\! \left(x \right)
F_{155}\! \left(x \right) = 0
F_{156}\! \left(x \right) = F_{157}\! \left(x \right)+F_{68}\! \left(x \right)
F_{157}\! \left(x \right) = F_{15}\! \left(x \right)+F_{158}\! \left(x \right)+F_{160}\! \left(x \right)+F_{173}\! \left(x \right)+F_{177}\! \left(x \right)
F_{158}\! \left(x \right) = F_{12}\! \left(x \right) F_{159}\! \left(x \right)
F_{159}\! \left(x \right) = F_{141}\! \left(x \right)+F_{157}\! \left(x \right)
F_{160}\! \left(x \right) = F_{12}\! \left(x \right) F_{161}\! \left(x \right)
F_{161}\! \left(x \right) = F_{162}\! \left(x \right)+F_{170}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right)+F_{166}\! \left(x \right)
F_{163}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{164}\! \left(x \right)+F_{40}\! \left(x \right)
F_{164}\! \left(x \right) = F_{12}\! \left(x \right) F_{165}\! \left(x \right)
F_{165}\! \left(x \right) = F_{145}\! \left(x \right)+F_{163}\! \left(x \right)
F_{166}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{167}\! \left(x \right)+F_{169}\! \left(x \right)
F_{167}\! \left(x \right) = F_{12}\! \left(x \right) F_{168}\! \left(x \right)
F_{168}\! \left(x \right) = F_{146}\! \left(x \right)+F_{166}\! \left(x \right)
F_{169}\! \left(x \right) = F_{12}\! \left(x \right) F_{162}\! \left(x \right)
F_{170}\! \left(x \right) = F_{171}\! \left(x \right)+F_{174}\! \left(x \right)
F_{171}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{172}\! \left(x \right)+F_{173}\! \left(x \right)
F_{172}\! \left(x \right) = F_{12}\! \left(x \right) F_{163}\! \left(x \right)
F_{173}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{174}\! \left(x \right) = 4 F_{15}\! \left(x \right)+F_{175}\! \left(x \right)+F_{176}\! \left(x \right)
F_{175}\! \left(x \right) = F_{12}\! \left(x \right) F_{166}\! \left(x \right)
F_{176}\! \left(x \right) = F_{12}\! \left(x \right) F_{170}\! \left(x \right)
F_{177}\! \left(x \right) = 0
F_{178}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \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_18(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_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_42(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_5(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_15(x) + F_27(x) + F_29(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_26(x) + F_6(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_36(x))
Eq(F_31(x), F_23(x) + F_32(x))
Eq(F_32(x), F_15(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_10(x) + F_32(x))
Eq(F_35(x), F_12(x)*F_31(x))
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_23(x))
Eq(F_39(x), 2*F_15(x) + F_40(x) + F_41(x))
Eq(F_40(x), F_12(x)*F_32(x))
Eq(F_41(x), F_12(x)*F_36(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_88(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_58(x))
Eq(F_45(x), F_46(x) + F_6(x))
Eq(F_46(x), F_15(x) + F_47(x) + F_56(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_10(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_49(x))
Eq(F_52(x), F_14(x) + F_53(x))
Eq(F_53(x), 2*F_15(x) + F_54(x) + F_55(x))
Eq(F_54(x), F_12(x)*F_50(x))
Eq(F_55(x), F_12(x)*F_52(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_13(x))
Eq(F_58(x), F_59(x) + F_72(x))
Eq(F_59(x), F_15(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_59(x) + F_6(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_31(x) + F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_15(x) + F_38(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_12(x) + F_65(x))
Eq(F_68(x), F_15(x) + F_40(x) + F_69(x) + F_71(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_14(x) + F_68(x))
Eq(F_71(x), F_12(x)*F_64(x))
Eq(F_72(x), F_15(x) + F_73(x) + F_75(x) + F_86(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_46(x) + F_72(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_82(x))
Eq(F_77(x), F_32(x) + F_78(x))
Eq(F_78(x), 2*F_15(x) + F_79(x) + F_81(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_50(x) + F_78(x))
Eq(F_81(x), F_12(x)*F_77(x))
Eq(F_82(x), F_39(x) + F_83(x))
Eq(F_83(x), 3*F_15(x) + F_84(x) + F_85(x))
Eq(F_84(x), F_12(x)*F_78(x))
Eq(F_85(x), F_12(x)*F_82(x))
Eq(F_86(x), F_12(x)*F_87(x))
Eq(F_87(x), F_64(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_101(x) + F_90(x))
Eq(F_90(x), F_19(x) + F_91(x))
Eq(F_91(x), F_100(x) + F_15(x) + F_92(x))
Eq(F_92(x), F_12(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_96(x))
Eq(F_94(x), F_10(x) + F_95(x))
Eq(F_95(x), F_15(x) + F_17(x) + F_47(x))
Eq(F_96(x), F_32(x) + F_97(x))
Eq(F_97(x), F_15(x) + F_71(x) + F_75(x) + F_98(x))
Eq(F_98(x), F_12(x)*F_99(x))
Eq(F_99(x), F_95(x) + F_97(x))
Eq(F_100(x), F_12(x)*F_90(x))
Eq(F_101(x), F_102(x) + F_137(x))
Eq(F_102(x), F_103(x) + F_15(x) + F_38(x))
Eq(F_103(x), F_104(x)*F_12(x))
Eq(F_104(x), F_105(x) + F_119(x))
Eq(F_105(x), F_106(x) + F_12(x))
Eq(F_106(x), F_107(x) + F_115(x) + F_15(x))
Eq(F_107(x), F_108(x)*F_12(x))
Eq(F_108(x), F_109(x) + F_112(x))
Eq(F_109(x), F_110(x) + F_12(x))
Eq(F_110(x), F_111(x))
Eq(F_111(x), F_109(x)*F_12(x))
Eq(F_112(x), F_113(x) + F_116(x))
Eq(F_113(x), F_114(x) + F_115(x) + F_15(x))
Eq(F_114(x), x**2)
Eq(F_115(x), x**2)
Eq(F_116(x), F_117(x) + F_118(x) + 2*F_15(x))
Eq(F_117(x), F_110(x)*F_12(x))
Eq(F_118(x), F_112(x)*F_12(x))
Eq(F_119(x), F_120(x) + F_65(x))
Eq(F_120(x), F_121(x) + F_123(x) + F_133(x) + F_15(x))
Eq(F_121(x), F_12(x)*F_122(x))
Eq(F_122(x), F_106(x) + F_120(x))
Eq(F_123(x), F_12(x)*F_124(x))
Eq(F_124(x), F_125(x) + F_130(x))
Eq(F_125(x), F_126(x) + F_65(x))
Eq(F_126(x), F_127(x) + F_129(x) + 2*F_15(x))
Eq(F_127(x), F_12(x)*F_128(x))
Eq(F_128(x), F_110(x) + F_126(x))
Eq(F_129(x), F_12(x)*F_125(x))
Eq(F_130(x), F_131(x) + F_134(x))
Eq(F_131(x), F_132(x) + F_133(x) + 2*F_15(x))
Eq(F_132(x), F_12(x)*F_65(x))
Eq(F_133(x), F_12(x)*F_37(x))
Eq(F_134(x), F_135(x) + F_136(x) + 3*F_15(x))
Eq(F_135(x), F_12(x)*F_126(x))
Eq(F_136(x), F_12(x)*F_130(x))
Eq(F_137(x), F_138(x) + F_15(x) + F_178(x) + F_40(x))
Eq(F_138(x), F_12(x)*F_139(x))
Eq(F_139(x), F_140(x) + F_156(x))
Eq(F_140(x), F_14(x) + F_141(x))
Eq(F_141(x), F_142(x) + F_15(x) + F_151(x) + F_155(x))
Eq(F_142(x), F_12(x)*F_143(x))
Eq(F_143(x), F_144(x) + F_148(x))
Eq(F_144(x), F_145(x) + F_146(x))
Eq(F_145(x), F_16(x))
Eq(F_146(x), F_147(x))
Eq(F_147(x), F_12(x)*F_144(x))
Eq(F_148(x), F_149(x) + F_152(x))
Eq(F_149(x), 2*F_15(x) + F_150(x) + F_151(x))
Eq(F_150(x), F_12(x)*F_145(x))
Eq(F_151(x), F_12(x)*F_14(x))
Eq(F_152(x), 3*F_15(x) + F_153(x) + F_154(x))
Eq(F_153(x), F_12(x)*F_146(x))
Eq(F_154(x), F_12(x)*F_148(x))
Eq(F_155(x), 0)
Eq(F_156(x), F_157(x) + F_68(x))
Eq(F_157(x), F_15(x) + F_158(x) + F_160(x) + F_173(x) + F_177(x))
Eq(F_158(x), F_12(x)*F_159(x))
Eq(F_159(x), F_141(x) + F_157(x))
Eq(F_160(x), F_12(x)*F_161(x))
Eq(F_161(x), F_162(x) + F_170(x))
Eq(F_162(x), F_163(x) + F_166(x))
Eq(F_163(x), 2*F_15(x) + F_164(x) + F_40(x))
Eq(F_164(x), F_12(x)*F_165(x))
Eq(F_165(x), F_145(x) + F_163(x))
Eq(F_166(x), 3*F_15(x) + F_167(x) + F_169(x))
Eq(F_167(x), F_12(x)*F_168(x))
Eq(F_168(x), F_146(x) + F_166(x))
Eq(F_169(x), F_12(x)*F_162(x))
Eq(F_170(x), F_171(x) + F_174(x))
Eq(F_171(x), 3*F_15(x) + F_172(x) + F_173(x))
Eq(F_172(x), F_12(x)*F_163(x))
Eq(F_173(x), F_12(x)*F_39(x))
Eq(F_174(x), 4*F_15(x) + F_175(x) + F_176(x))
Eq(F_175(x), F_12(x)*F_166(x))
Eq(F_176(x), F_12(x)*F_170(x))
Eq(F_177(x), 0)
Eq(F_178(x), F_101(x)*F_12(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:
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