0132_0213_0231_1023_3012
Counting sequence:
1, 1, 2, 6, 19, 58, 179, 566, 1840, 6132, 20862, 72175, 253123, 897721, 3213778, 11596823, 42133634, 153993254, 565778314, 2088366218, 7740501136, 28797385572, 107499037470, 402523260988, 1511461726642, 5690133569062, 21472291262882, 81205724395146, 307735510995538, 1168393454518598, 4443911107521282, 16929967108832707, 64597535794977168, 246833259265180849, 944456172281372811, 3618396346081818045, 13879548933581345346, 53300432498239495094, 204906557792936293110, 788544794038789834829, 3037524307580463962257, 11711572239180606102135, 45195308913324218420002, 174556598919442047820998, 674725640178339956274332, 2610060593072496959392013, 10103964906404484405558639, 39141485358610572963596187, 151730971910622596059385592, 588559887575298266255004278, 2284416603018852380698419072, 8871919557109230649831481189, 34475260007576579314843912021, 134040416745292537325704349443, 521426832461803956049026879278, 2029420210270894881026422362500, 7902484411872102628285526849534, 30786538426085300897350046413195, 119993161041559933780562972872019, 467890027979461038027186089807893, 1825224649995046277258301455009508, 7123082923308766350776774755228986, 27809488978337935469507781067403932, 108614003650983743815495772624698966, 424366581216395736548717308462395715, 1658647299359267242569948510772603446, 6485144233037003252390697084401473963, 25364913128754902264408450762937035523, 99240944952342970013949919622878869362, 388407843110153593999891084098377364623, 1520619923684847723216280183683505304235, 5955045908859645321855091939604902564920, 23328006652596695009667005676276286400679, 91410209630273229530572097356429522759535, 358288574584836146924391156949764225749952, 1404718011958296997148666100734614227134477, 5508840103928968780583445055479151810085251, 21609415232142729894385580303923491802919662, 84788083292886576588222954347048143558572299, 332761297810572943469152501823735302681713809, 1306274132909590837906287458662705375337782948, 5129048510150091949851970270895508677843843679, 20143631999947899534077934994761235359179320941, 79128851714760864032733865198009993588364111056, 310903610771197467993952114197676429458628808455, 1221822933462353749730281488378517531493765177551, 4802642363123671793589480180925856439090104570968, 18881636313793599912695805074586623587654880988083, 74247942544682596533892524844273532950075405035557, 292020075592297308762693314747832251978185092266268, 1148742329996493793163742467795994554144784606416555, 4519728754750707128525449400216738630094432400997035, 17786078771899170837095490000572991461443340426787484, 70004258446297015748523204384222019470327406502091695, 275577329181052858711559648812575861355245973417713437, 1085014932784374395880183494641413961490723497803459922, 4272672610412385150348881314489793791165379321162250831, 16828046747689703478120852429523298958056271175198510857, 66288250794494710416670708046555478647531352498848928274, 261160067674764385927614846502317269305123026325435433989, 1029065455957185651950330969232489277790774585591728457893
Generating function in Maple syntax:
1/2*((-3*x^4+x^3+3*x^2-3*x+1)*(1-4*x)^(1/2)-4*x^5+3*x^4+x^3-3*x^2+3*x-1)/x/(x^2+x-1)/(x-1)^2
Generating function in latex syntax:
\frac{\left(-3 x^{4}+x^{3}+3 x^{2}-3 x +1\right) \sqrt{1-4 x}-4 x^{5}+3 x^{4}+x^{3}-3 x^{2}+3 x -1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**5 + 3*x**4/2 + x**3/2 - 3*x**2/2 + 3*x/2 + sqrt(1 - 4*x)*(-3*x**4 + x**3 + 3*x**2 - 3*x + 1)/2 - 1/2)/(x*(x - 1)**2*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2+x-1)^2*(x-1)^4*F(x)^2+(x^2+x-1)*(4*x^5-3*x^4-x^3+3*x^2-3*x+1)*(x-1)^2*F(x)+4*x^9+3*x^8-8*x^7-8*x^6+18*x^5-4*x^4-13*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(x^{2}+x -1\right) \left(4 x^{5}-3 x^{4}-x^{3}+3 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{9}+3 x^{8}-8 x^{7}-8 x^{6}+18 x^{5}-4 x^{4}-13 x^{3}+14 x^{2}-6 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 179
a(7) = 566
a(8) = 1840
a(9) = 6132
a(n+8) = -6*(2*n+1)/(n+9)*a(n)+(4+7*n)/(n+9)*a(1+n)+(57+35*n)/(n+9)*a(n+2)-(41*n+104)/(n+9)*a(n+3)-(8*n+37)/(n+9)*a(n+4)+(213+40*n)/(n+9)*a(n+5)-(29*n+191)/(n+9)*a(n+6)+(70+9*n)/(n+9)*a(n+7)+2/(n+9), n >= 10
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) = 58
a \! \left(6\right) = 179
a \! \left(7\right) = 566
a \! \left(8\right) = 1840
a \! \left(9\right) = 6132
a \! \left(n +8\right) = -\frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +9}+\frac{\left(4+7 n \right) a \! \left(1+n \right)}{n +9}+\frac{\left(57+35 n \right) a \! \left(n +2\right)}{n +9}-\frac{\left(41 n +104\right) a \! \left(n +3\right)}{n +9}-\frac{\left(8 n +37\right) a \! \left(n +4\right)}{n +9}+\frac{\left(213+40 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(29 n +191\right) a \! \left(n +6\right)}{n +9}+\frac{\left(70+9 n \right) a \! \left(n +7\right)}{n +9}+\frac{2}{n +9}, \quad n \geq 10
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21560/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[21,x]*F[3,x]
F[3,x] = F[1,x]+F[104,x]+F[4,x]
F[4,x] = F[21,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[101,x,k[0]]+F[103,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]
F[8,x,k[0]] = F[21,x]*F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[99,x]
F[10,x,k[0]] = F[11,x,k[0]]+F[38,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]+F[16,x,k[0]]
F[12,x,k[0]] = F[1,x]+F[13,x,k[0]]
F[13,x,k[0]] = F[14,x,k[0]]
F[14,x,k[0]] = F[12,x,k[0]]*F[15,x,k[0]]
F[15,x,k[0]] = k[0]*x
F[16,x,k[0]] = F[17,x]+F[28,x,k[0]]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[21,x]
F[19,x] = F[20,x]+F[22,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = x
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[21,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[21,x]*F[23,x]
F[28,x,k[0]] = F[29,x]+F[30,x,k[0]]+F[35,x,k[0]]
F[29,x] = 0
F[30,x,k[0]] = F[21,x]*F[31,x,k[0]]
F[31,x,k[0]] = F[32,x,k[0]]+F[33,x,k[0]]
F[32,x,k[0]] = F[15,x,k[0]]
F[33,x,k[0]] = F[34,x,k[0]]
F[34,x,k[0]] = F[15,x,k[0]]*F[23,x]
F[35,x,k[0]] = F[15,x,k[0]]*F[36,x,k[0]]
F[36,x,k[0]] = F[23,x]+F[37,x,k[0]]
F[37,x,k[0]] = F[35,x,k[0]]
F[38,x,k[0]] = F[39,x,k[0]]+F[56,x,k[0]]
F[39,x,k[0]] = F[40,x]+F[50,x,k[0]]
F[40,x] = F[41,x]
F[41,x] = F[21,x]*F[42,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[1,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[21,x]*F[43,x]
F[46,x] = F[40,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[21,x]*F[49,x]
F[49,x] = F[40,x]
F[50,x,k[0]] = F[29,x]+F[51,x,k[0]]+F[55,x,k[0]]
F[51,x,k[0]] = F[21,x]*F[52,x,k[0]]
F[52,x,k[0]] = F[53,x,k[0]]+F[54,x,k[0]]
F[53,x,k[0]] = F[13,x,k[0]]
F[54,x,k[0]] = F[50,x,k[0]]
F[55,x,k[0]] = F[15,x,k[0]]*F[39,x,k[0]]
F[56,x,k[0]] = F[57,x]+F[79,x,k[0]]
F[57,x] = F[29,x]+F[58,x]+F[65,x]
F[58,x] = F[21,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[23,x]
F[61,x] = F[62,x]
F[62,x] = F[29,x]+F[58,x]+F[63,x]
F[63,x] = F[21,x]*F[64,x]
F[64,x] = F[40,x]+F[62,x]
F[65,x] = F[21,x]*F[66,x]
F[66,x] = F[67,x]+F[73,x]
F[67,x] = F[40,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[21,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[21,x]
F[72,x] = F[68,x]
F[73,x] = F[62,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[21,x]*F[76,x]
F[76,x] = F[77,x]+F[78,x]
F[77,x] = F[26,x]
F[78,x] = F[74,x]
F[79,x,k[0]] = F[29,x]+F[80,x,k[0]]+F[85,x,k[0]]+F[87,x,k[0]]
F[80,x,k[0]] = F[21,x]*F[81,x,k[0]]
F[81,x,k[0]] = F[82,x,k[0]]+F[83,x,k[0]]
F[82,x,k[0]] = F[37,x,k[0]]
F[83,x,k[0]] = F[84,x,k[0]]
F[84,x,k[0]] = 2*F[29,x]+F[80,x,k[0]]+F[85,x,k[0]]
F[85,x,k[0]] = F[15,x,k[0]]*F[86,x,k[0]]
F[86,x,k[0]] = F[62,x]+F[84,x,k[0]]
F[87,x,k[0]] = F[21,x]*F[88,x,k[0]]
F[88,x,k[0]] = F[89,x,k[0]]+F[94,x,k[0]]
F[89,x,k[0]] = F[90,x,k[0]]
F[90,x,k[0]] = F[21,x]*F[91,x,k[0]]
F[91,x,k[0]] = F[92,x,k[0]]+F[93,x,k[0]]
F[92,x,k[0]] = F[15,x,k[0]]
F[93,x,k[0]] = F[90,x,k[0]]
F[94,x,k[0]] = F[95,x,k[0]]
F[95,x,k[0]] = F[21,x]*F[96,x,k[0]]
F[96,x,k[0]] = F[97,x,k[0]]+F[98,x,k[0]]
F[97,x,k[0]] = F[34,x,k[0]]
F[98,x,k[0]] = F[95,x,k[0]]
F[99,x] = F[100,x]*F[21,x]
F[100,x] = F[40,x]+F[43,x]
F[101,x,k[0]] = F[102,x,k[0]]*F[21,x]
F[102,x,k[0]] = 1/(-1+k[0])*(F[6,x,k[0]]*k[0]-F[6,x,1])
F[103,x,k[0]] = F[15,x,k[0]]*F[6,x,k[0]]
F[104,x] = F[105,x]
F[105,x] = F[106,x]*F[21,x]
F[106,x] = F[107,x]+F[99,x]
F[107,x] = F[108,x]+F[109,x]
F[108,x] = F[1,x]+F[17,x]
F[109,x] = F[40,x]+F[57,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_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{104}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{101}\! \left(x , y\right)+F_{103}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{99}\! \left(x \right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{38}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = y x
F_{16}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{28}\! \left(x , y\right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = x
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_{21}\! \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_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x \right)+F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{31}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)+F_{33}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{34}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{23}\! \left(x \right)
F_{35}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{23}\! \left(x \right)+F_{37}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{35}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{39}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x \right)+F_{50}\! \left(x , y\right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{21}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{21}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)
F_{50}\! \left(x , y\right) = F_{29}\! \left(x \right)+F_{51}\! \left(x , y\right)+F_{55}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)+F_{54}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{13}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{50}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{39}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{57}\! \left(x \right)+F_{79}\! \left(x , y\right)
F_{57}\! \left(x \right) = F_{29}\! \left(x \right)+F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{23}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{29}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{40}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{67}\! \left(x \right) = F_{40}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{21}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{21}\! \left(x \right)
F_{72}\! \left(x \right) = F_{68}\! \left(x \right)
F_{73}\! \left(x \right) = F_{62}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{21}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{26}\! \left(x \right)
F_{78}\! \left(x \right) = F_{74}\! \left(x \right)
F_{79}\! \left(x , y\right) = F_{29}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{85}\! \left(x , y\right)+F_{87}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{81}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{82}\! \left(x , y\right)+F_{83}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{37}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = 2 F_{29}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{86}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{62}\! \left(x \right)+F_{84}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{88}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{89}\! \left(x , y\right)+F_{94}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{91}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{92}\! \left(x , y\right)+F_{93}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = F_{90}\! \left(x , y\right)
F_{94}\! \left(x , y\right) = F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{96}\! \left(x , y\right)
F_{96}\! \left(x , y\right) = F_{97}\! \left(x , y\right)+F_{98}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{34}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{95}\! \left(x , y\right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{21}\! \left(x \right)
F_{100}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{101}\! \left(x , y\right) = F_{102}\! \left(x , y\right) F_{21}\! \left(x \right)
F_{102}\! \left(x , y\right) = \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}
F_{103}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{6}\! \left(x , y\right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{21}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{99}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{109}\! \left(x \right)
F_{108}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{109}\! \left(x \right) = F_{40}\! \left(x \right)+F_{57}\! \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_21(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_104(x) + F_4(x))
Eq(F_4(x), F_21(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_101(x, k_0) + F_103(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0))
Eq(F_8(x, k_0), F_21(x)*F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_99(x))
Eq(F_10(x, k_0), F_11(x, k_0) + F_38(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0) + F_16(x, k_0))
Eq(F_12(x, k_0), F_1(x) + F_13(x, k_0))
Eq(F_13(x, k_0), F_14(x, k_0))
Eq(F_14(x, k_0), F_12(x, k_0)*F_15(x, k_0))
Eq(F_15(x, k_0), k_0*x)
Eq(F_16(x, k_0), F_17(x) + F_28(x, k_0))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_21(x))
Eq(F_19(x), F_20(x) + F_22(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), x)
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_21(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_21(x)*F_23(x))
Eq(F_28(x, k_0), F_29(x) + F_30(x, k_0) + F_35(x, k_0))
Eq(F_29(x), 0)
Eq(F_30(x, k_0), F_21(x)*F_31(x, k_0))
Eq(F_31(x, k_0), F_32(x, k_0) + F_33(x, k_0))
Eq(F_32(x, k_0), F_15(x, k_0))
Eq(F_33(x, k_0), F_34(x, k_0))
Eq(F_34(x, k_0), F_15(x, k_0)*F_23(x))
Eq(F_35(x, k_0), F_15(x, k_0)*F_36(x, k_0))
Eq(F_36(x, k_0), F_23(x) + F_37(x, k_0))
Eq(F_37(x, k_0), F_35(x, k_0))
Eq(F_38(x, k_0), F_39(x, k_0) + F_56(x, k_0))
Eq(F_39(x, k_0), F_40(x) + F_50(x, k_0))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_21(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_1(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_21(x)*F_43(x))
Eq(F_46(x), F_40(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_21(x)*F_49(x))
Eq(F_49(x), F_40(x))
Eq(F_50(x, k_0), F_29(x) + F_51(x, k_0) + F_55(x, k_0))
Eq(F_51(x, k_0), F_21(x)*F_52(x, k_0))
Eq(F_52(x, k_0), F_53(x, k_0) + F_54(x, k_0))
Eq(F_53(x, k_0), F_13(x, k_0))
Eq(F_54(x, k_0), F_50(x, k_0))
Eq(F_55(x, k_0), F_15(x, k_0)*F_39(x, k_0))
Eq(F_56(x, k_0), F_57(x) + F_79(x, k_0))
Eq(F_57(x), F_29(x) + F_58(x) + F_65(x))
Eq(F_58(x), F_21(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_23(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_29(x) + F_58(x) + F_63(x))
Eq(F_63(x), F_21(x)*F_64(x))
Eq(F_64(x), F_40(x) + F_62(x))
Eq(F_65(x), F_21(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_73(x))
Eq(F_67(x), F_40(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_21(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_21(x))
Eq(F_72(x), F_68(x))
Eq(F_73(x), F_62(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_21(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_78(x))
Eq(F_77(x), F_26(x))
Eq(F_78(x), F_74(x))
Eq(F_79(x, k_0), F_29(x) + F_80(x, k_0) + F_85(x, k_0) + F_87(x, k_0))
Eq(F_80(x, k_0), F_21(x)*F_81(x, k_0))
Eq(F_81(x, k_0), F_82(x, k_0) + F_83(x, k_0))
Eq(F_82(x, k_0), F_37(x, k_0))
Eq(F_83(x, k_0), F_84(x, k_0))
Eq(F_84(x, k_0), 2*F_29(x) + F_80(x, k_0) + F_85(x, k_0))
Eq(F_85(x, k_0), F_15(x, k_0)*F_86(x, k_0))
Eq(F_86(x, k_0), F_62(x) + F_84(x, k_0))
Eq(F_87(x, k_0), F_21(x)*F_88(x, k_0))
Eq(F_88(x, k_0), F_89(x, k_0) + F_94(x, k_0))
Eq(F_89(x, k_0), F_90(x, k_0))
Eq(F_90(x, k_0), F_21(x)*F_91(x, k_0))
Eq(F_91(x, k_0), F_92(x, k_0) + F_93(x, k_0))
Eq(F_92(x, k_0), F_15(x, k_0))
Eq(F_93(x, k_0), F_90(x, k_0))
Eq(F_94(x, k_0), F_95(x, k_0))
Eq(F_95(x, k_0), F_21(x)*F_96(x, k_0))
Eq(F_96(x, k_0), F_97(x, k_0) + F_98(x, k_0))
Eq(F_97(x, k_0), F_34(x, k_0))
Eq(F_98(x, k_0), F_95(x, k_0))
Eq(F_99(x), F_100(x)*F_21(x))
Eq(F_100(x), F_40(x) + F_43(x))
Eq(F_101(x, k_0), F_102(x, k_0)*F_21(x))
Eq(F_102(x, k_0), (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_103(x, k_0), F_15(x, k_0)*F_6(x, k_0))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_106(x)*F_21(x))
Eq(F_106(x), F_107(x) + F_99(x))
Eq(F_107(x), F_108(x) + F_109(x))
Eq(F_108(x), F_1(x) + F_17(x))
Eq(F_109(x), F_40(x) + F_57(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 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], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 0], [1, 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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": 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