0132_0213_0231_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 51, 117, 241, 464, 860, 1568, 2856, 5249, 9785, 18527, 35587, 69174, 135690, 267922, 531426, 1057295, 2107695, 4206937, 8403621, 16794924, 33575176, 67133012, 134245676, 268467629, 536907765, 1073783843, 2147531351, 4295021234, 8589995350, 17179937382, 34359814662, 68719561819, 137439048075, 274878011837, 549755929881, 1099511755720, 2199023396340, 4398046665672, 8796093191536, 17592186229529, 35184372290801, 70368744397607, 140737488594411, 281474976970094, 562949953702370, 1125899907146618, 2251799814013546, 4503599627724519, 9007199255122215, 18014398509891937, 36028797019404237, 72057594038400164, 144115188076361760, 288230376152253052, 576460752304002036, 1152921504607464645, 2305843009214352685, 4611686018428089707, 9223372036855522751, 18446744073710345834, 36893488147419946926, 73786976294839101902, 147573952589677362446, 295147905179353831859, 590295810358706716675, 1180591620717412429893, 2361183241434823797441, 4722366482869646471104, 9444732965739291754380, 18889465931478582254192, 37778931862957163184312, 75557863725914324972209, 151115727451828648472745, 302231454903657295395567, 604462909807314589159891, 1208925819614629176604070, 2417851639229258351404730, 4835703278458516700915042, 9671406556917033399841266, 19342813113834066797595839, 38685626227668133593003551, 77371252455336267183713897, 154742504910672534365025781, 309485009821345068727536924, 618970019642690137452442680, 1237940039285380274902133668, 2475880078570760549801391036, 4951760157141521099599776989, 9903520314283042199196415845, 19807040628566084398389556147, 39614081257132168796775694887, 79228162514264337593547825954, 158456325028528675187091937030, 316912650057057350374180003382, 633825300114114700748355975446, 1267650600228229401496707753995
Generating function in Maple syntax:
(2*x^7+x^6-8*x^5+12*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{7}+x^{6}-8 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**7 + x**6 - 8*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^5*F(x)+2*x^7+x^6-8*x^5+12*x^4-18*x^3+15*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}+x^{6}-8 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-6+2^n+1/24*n^4+5/12*n^3-85/24*n^2+85/12*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -6+2^{n}+\frac{n^{4}}{24}+\frac{5 n^{3}}{12}-\frac{85 n^{2}}{24}+\frac{85 n}{12} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 117
a(7) = 241
a(n+1) = -1/24*n^4-1/4*n^3+121/24*n^2+2*a(n)-51/4*n+10, 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 117
a \! \left(7\right) = 241
a \! \left(n +1\right) = -\frac{n^{4}}{24}-\frac{n^{3}}{4}+\frac{121 n^{2}}{24}+2 a \! \left(n \right)-\frac{51 n}{4}+10, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21594/
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[8,x]
F[4,x] = F[27,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = 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[8,x]
F[13,x] = F[14,x]
F[14,x] = F[10,x]*F[15,x]*F[8,x]
F[15,x] = F[16,x]+F[18,x]
F[16,x] = 2*F[17,x]+F[1,x]
F[17,x] = F[16,x]*F[8,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[22,x]
F[20,x] = 0
F[21,x] = F[18,x]*F[8,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[19,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[8,x]
F[26,x] = F[1,x]+F[24,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[139,x]+F[30,x]
F[30,x] = F[31,x]+F[67,x]
F[31,x] = F[32,x]+F[55,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[36,x]+F[43,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[8,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[24,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[40,x]*F[8,x]
F[43,x] = F[24,x]+F[44,x]
F[44,x] = F[20,x]+F[45,x]+F[47,x]
F[45,x] = F[46,x]*F[8,x]
F[46,x] = F[37,x]+F[44,x]
F[47,x] = F[48,x]*F[8,x]
F[48,x] = F[23,x]+F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]*F[8,x]
F[52,x] = F[24,x]+F[50,x]
F[53,x] = F[54,x]
F[54,x] = F[49,x]*F[8,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[58,x]*F[8,x]
F[58,x] = F[10,x]+F[23,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]*F[8,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[37,x]
F[63,x] = F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[66,x]*F[8,x]
F[66,x] = F[37,x]+F[64,x]
F[67,x] = F[108,x]+F[68,x]
F[68,x] = F[24,x]+F[69,x]
F[69,x] = F[20,x]+F[70,x]+F[90,x]
F[70,x] = F[71,x]*F[8,x]
F[71,x] = F[37,x]+F[72,x]
F[72,x] = F[20,x]+F[70,x]+F[73,x]
F[73,x] = F[74,x]*F[8,x]
F[74,x] = F[75,x]+F[79,x]
F[75,x] = F[24,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[78,x]*F[8,x]
F[78,x] = F[11,x]+F[76,x]
F[79,x] = F[80,x]+F[86,x]
F[80,x] = F[20,x]+F[81,x]+F[84,x]
F[81,x] = F[8,x]*F[82,x]
F[82,x] = F[24,x]+F[83,x]
F[83,x] = F[81,x]
F[84,x] = F[8,x]*F[85,x]
F[85,x] = F[24,x]+F[80,x]
F[86,x] = F[87,x]
F[87,x] = F[8,x]*F[88,x]
F[88,x] = F[41,x]+F[89,x]
F[89,x] = F[87,x]
F[90,x] = F[8,x]*F[91,x]
F[91,x] = F[92,x]+F[93,x]
F[92,x] = F[24,x]+F[64,x]
F[93,x] = F[80,x]+F[94,x]
F[94,x] = 2*F[20,x]+F[106,x]+F[95,x]
F[95,x] = F[8,x]*F[96,x]
F[96,x] = F[105,x]+F[97,x]
F[97,x] = F[100,x]+F[20,x]+F[98,x]
F[98,x] = F[8,x]*F[99,x]
F[99,x] = F[11,x]+F[41,x]
F[100,x] = F[101,x]*F[8,x]
F[101,x] = F[102,x]+F[40,x]
F[102,x] = F[103,x]+F[50,x]
F[103,x] = F[104,x]
F[104,x] = F[102,x]*F[8,x]
F[105,x] = F[95,x]
F[106,x] = F[107,x]*F[8,x]
F[107,x] = F[64,x]+F[94,x]
F[108,x] = F[109,x]+F[131,x]
F[109,x] = F[110,x]+F[114,x]+F[20,x]
F[110,x] = F[111,x]*F[8,x]
F[111,x] = F[11,x]+F[112,x]
F[112,x] = F[110,x]+F[113,x]+F[20,x]
F[113,x] = F[75,x]*F[8,x]
F[114,x] = F[115,x]*F[8,x]
F[115,x] = F[116,x]+F[75,x]
F[116,x] = F[117,x]+F[123,x]
F[117,x] = F[118,x]+F[121,x]+F[20,x]
F[118,x] = F[119,x]*F[8,x]
F[119,x] = F[120,x]+F[8,x]
F[120,x] = F[118,x]
F[121,x] = F[122,x]*F[8,x]
F[122,x] = F[117,x]+F[24,x]
F[123,x] = 2*F[20,x]+F[124,x]+F[129,x]
F[124,x] = F[125,x]*F[8,x]
F[125,x] = F[126,x]+F[128,x]
F[126,x] = F[127,x]
F[127,x] = F[11,x]*F[8,x]
F[128,x] = F[124,x]
F[129,x] = F[130,x]*F[8,x]
F[130,x] = F[123,x]+F[76,x]
F[131,x] = F[132,x]
F[132,x] = F[133,x]*F[8,x]
F[133,x] = F[134,x]+F[135,x]
F[134,x] = F[64,x]
F[135,x] = F[136,x]
F[136,x] = F[137,x]
F[137,x] = F[138,x]*F[8,x]
F[138,x] = F[136,x]+F[64,x]
F[139,x] = F[140,x]
F[140,x] = F[141,x]*F[26,x]
F[141,x] = F[142,x]+F[143,x]
F[142,x] = F[24,x]*F[26,x]
F[143,x] = F[144,x]
F[144,x] = F[16,x]*F[24,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{27}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{1}\! \left(x \right)
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \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_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{8}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{139}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{55}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{43}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{8}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{20}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right) F_{8}\! \left(x \right)
F_{46}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right) F_{8}\! \left(x \right)
F_{52}\! \left(x \right) = F_{24}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right) F_{8}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right) F_{8}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right) F_{8}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{37}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right) F_{8}\! \left(x \right)
F_{66}\! \left(x \right) = F_{37}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{108}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{24}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{20}\! \left(x \right)+F_{70}\! \left(x \right)+F_{90}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right) F_{8}\! \left(x \right)
F_{71}\! \left(x \right) = F_{37}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{20}\! \left(x \right)+F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right) F_{8}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{79}\! \left(x \right)
F_{75}\! \left(x \right) = F_{24}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right) F_{8}\! \left(x \right)
F_{78}\! \left(x \right) = F_{11}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{86}\! \left(x \right)
F_{80}\! \left(x \right) = F_{20}\! \left(x \right)+F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{81}\! \left(x \right) = F_{8}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{24}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{81}\! \left(x \right)
F_{84}\! \left(x \right) = F_{8}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{24}\! \left(x \right)+F_{80}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{8}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{41}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{8}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{93}\! \left(x \right)
F_{92}\! \left(x \right) = F_{24}\! \left(x \right)+F_{64}\! \left(x \right)
F_{93}\! \left(x \right) = F_{80}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{106}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{8}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{105}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{100}\! \left(x \right)+F_{20}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{8}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{11}\! \left(x \right)+F_{41}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{8}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{40}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{50}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{102}\! \left(x \right) F_{8}\! \left(x \right)
F_{105}\! \left(x \right) = F_{95}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{8}\! \left(x \right)
F_{107}\! \left(x \right) = F_{64}\! \left(x \right)+F_{94}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{131}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{114}\! \left(x \right)+F_{20}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{8}\! \left(x \right)
F_{111}\! \left(x \right) = F_{11}\! \left(x \right)+F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right)+F_{113}\! \left(x \right)+F_{20}\! \left(x \right)
F_{113}\! \left(x \right) = F_{75}\! \left(x \right) F_{8}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{8}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{75}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{123}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{121}\! \left(x \right)+F_{20}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{8}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{8}\! \left(x \right)
F_{120}\! \left(x \right) = F_{118}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{8}\! \left(x \right)
F_{122}\! \left(x \right) = F_{117}\! \left(x \right)+F_{24}\! \left(x \right)
F_{123}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{124}\! \left(x \right)+F_{129}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{8}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{128}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{128}\! \left(x \right) = F_{124}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{8}\! \left(x \right)
F_{130}\! \left(x \right) = F_{123}\! \left(x \right)+F_{76}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right) F_{8}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)+F_{135}\! \left(x \right)
F_{134}\! \left(x \right) = F_{64}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{8}\! \left(x \right)
F_{138}\! \left(x \right) = F_{136}\! \left(x \right)+F_{64}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right) F_{26}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{143}\! \left(x \right)
F_{142}\! \left(x \right) = F_{24}\! \left(x \right) F_{26}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{16}\! \left(x \right) F_{24}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_27(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_8(x)*F_9(x))
Eq(F_8(x), 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_8(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_10(x)*F_15(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_18(x))
Eq(F_16(x), F_1(x) + 2*F_17(x))
Eq(F_17(x), F_16(x)*F_8(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_22(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_18(x)*F_8(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_19(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_8(x))
Eq(F_26(x), F_1(x) + F_24(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_139(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_67(x))
Eq(F_31(x), F_32(x) + F_55(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_36(x) + F_43(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_8(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_24(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_40(x)*F_8(x))
Eq(F_43(x), F_24(x) + F_44(x))
Eq(F_44(x), F_20(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_46(x)*F_8(x))
Eq(F_46(x), F_37(x) + F_44(x))
Eq(F_47(x), F_48(x)*F_8(x))
Eq(F_48(x), F_23(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x)*F_8(x))
Eq(F_52(x), F_24(x) + F_50(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_49(x)*F_8(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_58(x)*F_8(x))
Eq(F_58(x), F_10(x) + F_23(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_61(x)*F_8(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_37(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_66(x)*F_8(x))
Eq(F_66(x), F_37(x) + F_64(x))
Eq(F_67(x), F_108(x) + F_68(x))
Eq(F_68(x), F_24(x) + F_69(x))
Eq(F_69(x), F_20(x) + F_70(x) + F_90(x))
Eq(F_70(x), F_71(x)*F_8(x))
Eq(F_71(x), F_37(x) + F_72(x))
Eq(F_72(x), F_20(x) + F_70(x) + F_73(x))
Eq(F_73(x), F_74(x)*F_8(x))
Eq(F_74(x), F_75(x) + F_79(x))
Eq(F_75(x), F_24(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_78(x)*F_8(x))
Eq(F_78(x), F_11(x) + F_76(x))
Eq(F_79(x), F_80(x) + F_86(x))
Eq(F_80(x), F_20(x) + F_81(x) + F_84(x))
Eq(F_81(x), F_8(x)*F_82(x))
Eq(F_82(x), F_24(x) + F_83(x))
Eq(F_83(x), F_81(x))
Eq(F_84(x), F_8(x)*F_85(x))
Eq(F_85(x), F_24(x) + F_80(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_8(x)*F_88(x))
Eq(F_88(x), F_41(x) + F_89(x))
Eq(F_89(x), F_87(x))
Eq(F_90(x), F_8(x)*F_91(x))
Eq(F_91(x), F_92(x) + F_93(x))
Eq(F_92(x), F_24(x) + F_64(x))
Eq(F_93(x), F_80(x) + F_94(x))
Eq(F_94(x), F_106(x) + 2*F_20(x) + F_95(x))
Eq(F_95(x), F_8(x)*F_96(x))
Eq(F_96(x), F_105(x) + F_97(x))
Eq(F_97(x), F_100(x) + F_20(x) + F_98(x))
Eq(F_98(x), F_8(x)*F_99(x))
Eq(F_99(x), F_11(x) + F_41(x))
Eq(F_100(x), F_101(x)*F_8(x))
Eq(F_101(x), F_102(x) + F_40(x))
Eq(F_102(x), F_103(x) + F_50(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_102(x)*F_8(x))
Eq(F_105(x), F_95(x))
Eq(F_106(x), F_107(x)*F_8(x))
Eq(F_107(x), F_64(x) + F_94(x))
Eq(F_108(x), F_109(x) + F_131(x))
Eq(F_109(x), F_110(x) + F_114(x) + F_20(x))
Eq(F_110(x), F_111(x)*F_8(x))
Eq(F_111(x), F_11(x) + F_112(x))
Eq(F_112(x), F_110(x) + F_113(x) + F_20(x))
Eq(F_113(x), F_75(x)*F_8(x))
Eq(F_114(x), F_115(x)*F_8(x))
Eq(F_115(x), F_116(x) + F_75(x))
Eq(F_116(x), F_117(x) + F_123(x))
Eq(F_117(x), F_118(x) + F_121(x) + F_20(x))
Eq(F_118(x), F_119(x)*F_8(x))
Eq(F_119(x), F_120(x) + F_8(x))
Eq(F_120(x), F_118(x))
Eq(F_121(x), F_122(x)*F_8(x))
Eq(F_122(x), F_117(x) + F_24(x))
Eq(F_123(x), F_124(x) + F_129(x) + 2*F_20(x))
Eq(F_124(x), F_125(x)*F_8(x))
Eq(F_125(x), F_126(x) + F_128(x))
Eq(F_126(x), F_127(x))
Eq(F_127(x), F_11(x)*F_8(x))
Eq(F_128(x), F_124(x))
Eq(F_129(x), F_130(x)*F_8(x))
Eq(F_130(x), F_123(x) + F_76(x))
Eq(F_131(x), F_132(x))
Eq(F_132(x), F_133(x)*F_8(x))
Eq(F_133(x), F_134(x) + F_135(x))
Eq(F_134(x), F_64(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_137(x))
Eq(F_137(x), F_138(x)*F_8(x))
Eq(F_138(x), F_136(x) + F_64(x))
Eq(F_139(x), F_140(x))
Eq(F_140(x), F_141(x)*F_26(x))
Eq(F_141(x), F_142(x) + F_143(x))
Eq(F_142(x), F_24(x)*F_26(x))
Eq(F_143(x), F_144(x))
Eq(F_144(x), F_16(x)*F_24(x)*F_8(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": 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