0123_1230_2310
Counting sequence:
1, 1, 2, 6, 21, 75, 248, 735, 1952, 4697, 10378, 21320, 41163, 75363, 131808, 221561, 359742, 566561, 868514, 1299754, 1903649, 2734539, 3859704, 5361555, 7340060, 9915417, 13230986, 17456492, 22791511, 29469251, 37760640, 47978733, 60483450, 75686657, 94057602, 116128718, 142501805, 173854603, 210947768, 254632263, 305857176, 365677977, 435265226, 515913744, 609052259, 716253539, 839245024, 979919969, 1140349110, 1322792865, 1529714082, 1763791346, 2027932857, 2325290891, 2659276856, 3033576955, 3452168468, 3919336665, 4439692362, 5018190132, 5660147183, 6371262915, 7157639168, 8025801173, 8982719218, 10035831041, 11193064962, 12462863766, 13854209349, 15376648139, 17040317304, 18855971759, 20835011984, 22989512665, 25332252170, 27876742872, 30637262331, 33628885347, 36867516896, 40369925961, 44153780270, 48237681953, 52641204130, 57384928442, 62490483537, 67980584523, 73879073400, 80210960483, 87002466828, 94281067673, 102075536906, 110415992572, 119333943431, 128862336579, 139035606144, 149889723069, 161462245994, 173792373249, 186920995970, 200890752350, 215746083037
Generating function in Maple syntax:
-(x^8-3*x^7-12*x^6+23*x^5-43*x^4+38*x^3-22*x^2+7*x-1)/(x-1)^8
Generating function in latex syntax:
-\frac{x^{8}-3 x^{7}-12 x^{6}+23 x^{5}-43 x^{4}+38 x^{3}-22 x^{2}+7 x -1}{\left(x -1\right)^{8}}
Generating function in sympy syntax:
(-x**8 + 3*x**7 + 12*x**6 - 23*x**5 + 43*x**4 - 38*x**3 + 22*x**2 - 7*x + 1)/(x - 1)**8
Implicit equation for the generating function in Maple syntax:
(x-1)^8*F(x)+x^8-3*x^7-12*x^6+23*x^5-43*x^4+38*x^3-22*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{8} F \! \left(x \right)+x^{8}-3 x^{7}-12 x^{6}+23 x^{5}-43 x^{4}+38 x^{3}-22 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,2-953/420*n-43/144*n^4+11/80*n^3+31/240*n^5-17/720*n^6+1/420*n^7+119/90*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 2-\frac{953}{420} n -\frac{43}{144} n^{4}+\frac{11}{80} n^{3}+\frac{31}{240} n^{5}-\frac{17}{720} n^{6}+\frac{1}{420} n^{7}+\frac{119}{90} n^{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 248
a(7) = 735
a(8) = 1952
a(n) = 1/5040*(n+2)*(12*n^6-143*n^5+937*n^4-3379*n^3+7451*n^2-8238*n+5040), 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) = 21
a \! \left(5\right) = 75
a \! \left(6\right) = 248
a \! \left(7\right) = 735
a \! \left(8\right) = 1952
a \! \left(n \right) = \frac{\left(n +2\right) \left(12 n^{6}-143 n^{5}+937 n^{4}-3379 n^{3}+7451 n^{2}-8238 n +5040\right)}{5040}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14668/
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[45,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[7,x]
F[14,x] = F[15,x]+F[16,x]+F[37,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[23,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[4,x]
F[22,x] = F[20,x]
F[23,x] = F[14,x]+F[24,x]
F[24,x] = 2*F[15,x]+F[25,x]+F[29,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[19,x]
F[28,x] = F[24,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[4,x]
F[34,x] = 2*F[15,x]+F[29,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[22,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[1,x]+F[4,x]
F[42,x] = F[15,x]+F[37,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]+F[22,x]
F[45,x] = F[2,x]+F[46,x]
F[46,x] = F[15,x]+F[160,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[74,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[15,x]+F[16,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[57,x]
F[53,x] = F[39,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[11,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[15,x]+F[43,x]+F[51,x]
F[59,x] = 2*F[15,x]+F[60,x]+F[62,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[19,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[68,x]
F[64,x] = F[54,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[19,x]
F[68,x] = F[59,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[73,x]
F[72,x] = F[65,x]
F[73,x] = F[69,x]
F[74,x] = F[46,x]+F[75,x]
F[75,x] = F[140,x]+F[15,x]+F[150,x]+F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[86,x]
F[78,x] = F[50,x]+F[79,x]
F[79,x] = F[15,x]+F[25,x]+F[62,x]+F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]+F[84,x]
F[82,x] = F[83,x]
F[83,x] = F[32,x]
F[84,x] = F[85,x]
F[85,x] = 2*F[15,x]+F[35,x]+F[80,x]
F[86,x] = F[75,x]+F[87,x]
F[87,x] = F[111,x]+F[121,x]+F[131,x]+F[15,x]+F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[90,x]+F[92,x]
F[90,x] = F[79,x]+F[91,x]
F[91,x] = F[70,x]
F[92,x] = F[87,x]+F[93,x]
F[93,x] = F[109,x]+F[110,x]+F[15,x]+F[94,x]+F[98,x]+F[99,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[96,x]+F[97,x]
F[96,x] = F[91,x]
F[97,x] = F[93,x]
F[98,x] = 0
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[105,x]
F[101,x] = F[102,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[65,x]
F[105,x] = F[106,x]
F[106,x] = 4*F[15,x]+F[107,x]+F[99,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[69,x]
F[109,x] = 0
F[110,x] = 0
F[111,x] = F[112,x]*F[4,x]
F[112,x] = F[113,x]+F[117,x]
F[113,x] = F[114,x]
F[114,x] = F[115,x]
F[115,x] = F[116,x]*F[4,x]
F[116,x] = F[19,x]+F[65,x]
F[117,x] = F[118,x]
F[118,x] = 3*F[15,x]+F[111,x]+F[119,x]
F[119,x] = F[120,x]*F[4,x]
F[120,x] = F[24,x]+F[69,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[123,x]+F[127,x]
F[123,x] = F[102,x]+F[124,x]
F[124,x] = F[125,x]
F[125,x] = F[126,x]*F[4,x]
F[126,x] = F[54,x]
F[127,x] = F[106,x]+F[128,x]
F[128,x] = 3*F[15,x]+F[121,x]+F[129,x]
F[129,x] = F[130,x]*F[4,x]
F[130,x] = F[59,x]
F[131,x] = F[132,x]*F[4,x]
F[132,x] = F[133,x]+F[136,x]
F[133,x] = F[134,x]
F[134,x] = F[135,x]*F[4,x]
F[135,x] = F[83,x]
F[136,x] = F[137,x]
F[137,x] = 3*F[15,x]+F[131,x]+F[138,x]
F[138,x] = F[139,x]*F[4,x]
F[139,x] = F[85,x]
F[140,x] = F[141,x]*F[4,x]
F[141,x] = F[142,x]+F[146,x]
F[142,x] = F[114,x]+F[143,x]
F[143,x] = F[144,x]
F[144,x] = F[145,x]*F[4,x]
F[145,x] = F[11,x]+F[54,x]
F[146,x] = F[118,x]+F[147,x]
F[147,x] = 2*F[15,x]+F[140,x]+F[148,x]
F[148,x] = F[149,x]*F[4,x]
F[149,x] = F[14,x]+F[59,x]
F[150,x] = F[151,x]*F[4,x]
F[151,x] = F[152,x]+F[156,x]
F[152,x] = F[124,x]+F[153,x]
F[153,x] = F[154,x]
F[154,x] = F[155,x]*F[4,x]
F[155,x] = F[39,x]+F[83,x]
F[156,x] = F[128,x]+F[157,x]
F[157,x] = 2*F[15,x]+F[150,x]+F[158,x]
F[158,x] = F[159,x]*F[4,x]
F[159,x] = F[58,x]+F[85,x]
F[160,x] = F[161,x]*F[4,x]
F[161,x] = F[162,x]+F[166,x]
F[162,x] = F[143,x]+F[163,x]
F[163,x] = F[164,x]
F[164,x] = F[165,x]*F[4,x]
F[165,x] = F[1,x]+F[39,x]
F[166,x] = F[147,x]+F[167,x]
F[167,x] = F[15,x]+F[160,x]+F[168,x]
F[168,x] = F[169,x]*F[4,x]
F[169,x] = F[58,x]+F[7,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_{45}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{37}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \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_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \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_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \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_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{160}\! \left(x \right)+F_{47}\! \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_{74}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{11}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{19}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{54}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right)
F_{68}\! \left(x \right) = F_{59}\! \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_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{65}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{46}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{140}\! \left(x \right)+F_{15}\! \left(x \right)+F_{150}\! \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_{86}\! \left(x \right)
F_{78}\! \left(x \right) = F_{50}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{62}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{84}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{32}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{80}\! \left(x \right)
F_{86}\! \left(x \right) = F_{75}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{111}\! \left(x \right)+F_{121}\! \left(x \right)+F_{131}\! \left(x \right)+F_{15}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{92}\! \left(x \right)
F_{90}\! \left(x \right) = F_{79}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{70}\! \left(x \right)
F_{92}\! \left(x \right) = F_{87}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)+F_{15}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)+F_{99}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{97}\! \left(x \right)
F_{96}\! \left(x \right) = F_{91}\! \left(x \right)
F_{97}\! \left(x \right) = F_{93}\! \left(x \right)
F_{98}\! \left(x \right) = 0
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{105}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{65}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = 4 F_{15}\! \left(x \right)+F_{107}\! \left(x \right)+F_{99}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{69}\! \left(x \right)
F_{109}\! \left(x \right) = 0
F_{110}\! \left(x \right) = 0
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{4}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{117}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{4}\! \left(x \right)
F_{116}\! \left(x \right) = F_{19}\! \left(x \right)+F_{65}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{111}\! \left(x \right)+F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right) F_{4}\! \left(x \right)
F_{120}\! \left(x \right) = F_{24}\! \left(x \right)+F_{69}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{127}\! \left(x \right)
F_{123}\! \left(x \right) = F_{102}\! \left(x \right)+F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{54}\! \left(x \right)
F_{127}\! \left(x \right) = F_{106}\! \left(x \right)+F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{121}\! \left(x \right)+F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{4}\! \left(x \right)
F_{130}\! \left(x \right) = F_{59}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{4}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{136}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right) F_{4}\! \left(x \right)
F_{135}\! \left(x \right) = F_{83}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{131}\! \left(x \right)+F_{138}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right) F_{4}\! \left(x \right)
F_{139}\! \left(x \right) = F_{85}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right) F_{4}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{146}\! \left(x \right)
F_{142}\! \left(x \right) = F_{114}\! \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_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{146}\! \left(x \right) = F_{118}\! \left(x \right)+F_{147}\! \left(x \right)
F_{147}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{140}\! \left(x \right)+F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right) F_{4}\! \left(x \right)
F_{149}\! \left(x \right) = F_{14}\! \left(x \right)+F_{59}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right) F_{4}\! \left(x \right)
F_{151}\! \left(x \right) = F_{152}\! \left(x \right)+F_{156}\! \left(x \right)
F_{152}\! \left(x \right) = F_{124}\! \left(x \right)+F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right) F_{4}\! \left(x \right)
F_{155}\! \left(x \right) = F_{39}\! \left(x \right)+F_{83}\! \left(x \right)
F_{156}\! \left(x \right) = F_{128}\! \left(x \right)+F_{157}\! \left(x \right)
F_{157}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{150}\! \left(x \right)+F_{158}\! \left(x \right)
F_{158}\! \left(x \right) = F_{159}\! \left(x \right) F_{4}\! \left(x \right)
F_{159}\! \left(x \right) = F_{58}\! \left(x \right)+F_{85}\! \left(x \right)
F_{160}\! \left(x \right) = F_{161}\! \left(x \right) F_{4}\! \left(x \right)
F_{161}\! \left(x \right) = F_{162}\! \left(x \right)+F_{166}\! \left(x \right)
F_{162}\! \left(x \right) = F_{143}\! \left(x \right)+F_{163}\! \left(x \right)
F_{163}\! \left(x \right) = F_{164}\! \left(x \right)
F_{164}\! \left(x \right) = F_{165}\! \left(x \right) F_{4}\! \left(x \right)
F_{165}\! \left(x \right) = F_{1}\! \left(x \right)+F_{39}\! \left(x \right)
F_{166}\! \left(x \right) = F_{147}\! \left(x \right)+F_{167}\! \left(x \right)
F_{167}\! \left(x \right) = F_{15}\! \left(x \right)+F_{160}\! \left(x \right)+F_{168}\! \left(x \right)
F_{168}\! \left(x \right) = F_{169}\! \left(x \right) F_{4}\! \left(x \right)
F_{169}\! \left(x \right) = F_{58}\! \left(x \right)+F_{7}\! \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_45(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_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_37(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_23(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_4(x))
Eq(F_22(x), F_20(x))
Eq(F_23(x), F_14(x) + F_24(x))
Eq(F_24(x), 2*F_15(x) + F_25(x) + F_29(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_19(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_4(x))
Eq(F_34(x), 2*F_15(x) + F_29(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_22(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_1(x) + F_4(x))
Eq(F_42(x), F_15(x) + F_37(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x) + F_22(x))
Eq(F_45(x), F_2(x) + F_46(x))
Eq(F_46(x), F_15(x) + F_160(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_74(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_15(x) + F_16(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_57(x))
Eq(F_53(x), F_39(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_11(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_15(x) + F_43(x) + F_51(x))
Eq(F_59(x), 2*F_15(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_19(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_68(x))
Eq(F_64(x), F_54(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_19(x))
Eq(F_68(x), F_59(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) + F_73(x))
Eq(F_72(x), F_65(x))
Eq(F_73(x), F_69(x))
Eq(F_74(x), F_46(x) + F_75(x))
Eq(F_75(x), F_140(x) + F_15(x) + F_150(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_86(x))
Eq(F_78(x), F_50(x) + F_79(x))
Eq(F_79(x), F_15(x) + F_25(x) + F_62(x) + F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_84(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_32(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), 2*F_15(x) + F_35(x) + F_80(x))
Eq(F_86(x), F_75(x) + F_87(x))
Eq(F_87(x), F_111(x) + F_121(x) + F_131(x) + F_15(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_90(x) + F_92(x))
Eq(F_90(x), F_79(x) + F_91(x))
Eq(F_91(x), F_70(x))
Eq(F_92(x), F_87(x) + F_93(x))
Eq(F_93(x), F_109(x) + F_110(x) + F_15(x) + F_94(x) + F_98(x) + F_99(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_97(x))
Eq(F_96(x), F_91(x))
Eq(F_97(x), F_93(x))
Eq(F_98(x), 0)
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_105(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_65(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_107(x) + 4*F_15(x) + F_99(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_69(x))
Eq(F_109(x), 0)
Eq(F_110(x), 0)
Eq(F_111(x), F_112(x)*F_4(x))
Eq(F_112(x), F_113(x) + F_117(x))
Eq(F_113(x), F_114(x))
Eq(F_114(x), F_115(x))
Eq(F_115(x), F_116(x)*F_4(x))
Eq(F_116(x), F_19(x) + F_65(x))
Eq(F_117(x), F_118(x))
Eq(F_118(x), F_111(x) + F_119(x) + 3*F_15(x))
Eq(F_119(x), F_120(x)*F_4(x))
Eq(F_120(x), F_24(x) + F_69(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_123(x) + F_127(x))
Eq(F_123(x), F_102(x) + F_124(x))
Eq(F_124(x), F_125(x))
Eq(F_125(x), F_126(x)*F_4(x))
Eq(F_126(x), F_54(x))
Eq(F_127(x), F_106(x) + F_128(x))
Eq(F_128(x), F_121(x) + F_129(x) + 3*F_15(x))
Eq(F_129(x), F_130(x)*F_4(x))
Eq(F_130(x), F_59(x))
Eq(F_131(x), F_132(x)*F_4(x))
Eq(F_132(x), F_133(x) + F_136(x))
Eq(F_133(x), F_134(x))
Eq(F_134(x), F_135(x)*F_4(x))
Eq(F_135(x), F_83(x))
Eq(F_136(x), F_137(x))
Eq(F_137(x), F_131(x) + F_138(x) + 3*F_15(x))
Eq(F_138(x), F_139(x)*F_4(x))
Eq(F_139(x), F_85(x))
Eq(F_140(x), F_141(x)*F_4(x))
Eq(F_141(x), F_142(x) + F_146(x))
Eq(F_142(x), F_114(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_11(x) + F_54(x))
Eq(F_146(x), F_118(x) + F_147(x))
Eq(F_147(x), F_140(x) + F_148(x) + 2*F_15(x))
Eq(F_148(x), F_149(x)*F_4(x))
Eq(F_149(x), F_14(x) + F_59(x))
Eq(F_150(x), F_151(x)*F_4(x))
Eq(F_151(x), F_152(x) + F_156(x))
Eq(F_152(x), F_124(x) + F_153(x))
Eq(F_153(x), F_154(x))
Eq(F_154(x), F_155(x)*F_4(x))
Eq(F_155(x), F_39(x) + F_83(x))
Eq(F_156(x), F_128(x) + F_157(x))
Eq(F_157(x), 2*F_15(x) + F_150(x) + F_158(x))
Eq(F_158(x), F_159(x)*F_4(x))
Eq(F_159(x), F_58(x) + F_85(x))
Eq(F_160(x), F_161(x)*F_4(x))
Eq(F_161(x), F_162(x) + F_166(x))
Eq(F_162(x), F_143(x) + F_163(x))
Eq(F_163(x), F_164(x))
Eq(F_164(x), F_165(x)*F_4(x))
Eq(F_165(x), F_1(x) + F_39(x))
Eq(F_166(x), F_147(x) + F_167(x))
Eq(F_167(x), F_15(x) + F_160(x) + F_168(x))
Eq(F_168(x), F_169(x)*F_4(x))
Eq(F_169(x), F_58(x) + F_7(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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": 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