0123_0213_0231_1230_2013
Counting sequence:
1, 1, 2, 6, 19, 60, 191, 619, 2048, 6909, 23704, 82489, 290500, 1033399, 3707838, 13402682, 48760351, 178405140, 656043839, 2423307028, 8987427447, 33453694466, 124936258105, 467995871754, 1757900019077, 6619846420528, 24987199492679, 94520750408682, 358268702159041, 1360510918810408, 5175497420902711, 19720133460129619, 75254198337177816, 287590328749420925, 1100534370899151688, 4216819865806452949, 16176618251666906440, 62127422576288648803, 238861285363295350202, 919286657093271150591, 3541413699369763259410, 13655332291007661393429, 52699762202912105352668, 203553241407997457013367, 786853361000994150101406, 3043971215078242223355125, 11784299926611415613401444, 45653073683802462499830933, 176980971925971827977822832, 686533217105588966032431403, 2664794874861749619656205858, 10349580545376066004887022013, 40218747491148691955029439524, 156376618946931126205583285403, 608336336974884597653192794826, 2367750953583703468645672670797, 9220207881428576966195331135108, 35921160738203428870440552047771, 140009621027325732903938870859850, 545954616154902718634582314226961, 2129805580751022761321355093265856, 8311933539335878411808435940482191, 32451671282381505237519894486755502, 126747521841153485025455279433135626, 495226691716970144504464322146682575, 1935645264867889813376954216390547924, 7568326849428202548370869922239693023, 29602051871384720011729422536296642972, 115820975870345005737915063199997751471, 453306478380560981294698856655056376170, 1774728586800843251784641033845285920769, 6950298511446948811203581227840351637118, 27227188901156348674132579796094992662817, 106690678266233726515340817428444261547316, 418187556577337047652877108947253395574555, 1639583211007715859481637830955215526365574, 6429992141371019770810024039350079987390093, 25223134868180904346021385318437625180640132, 98968378479713362805711537173084954419975731, 388418459654978262259995383202575721684367960, 1524778037602314534191628260207243178351981899, 5987068087590635016654869557713376361851636638, 23513653103207411851390010075629031998247870917, 92368094235987606559278062110297679141233077016, 362925545275382724587920525399466245561904357455, 1426279248197656560560957183442942704285007761974, 5606359321754181295075653011613850162713759076293, 22041674156419608092144797972376736306081349471232, 86674934742182522462641435458522917768762884732231, 340899093046183318986594876237364564855310256758830, 1341033693846538100915994126773906429217771346709629, 5276345927430542786333847699537255938991803026733428, 20763703749922432193462174663315599171005734154569027, 81724580285262847944924738244145248062975463062007026, 321717925803339853113840514867621971069256290550236625, 1266691723780768060966446385322561567906487048785140674, 4988134928186722446530316926701808227615993746163834973, 19646064284316297883547194773358840989328948697063590072, 77389422353917655665734895381401697323349680321820201071, 304898253148147005327554435777090551279391362923361248410, 1201418200238278502014724505851191183700228884462107157729
Generating function in Maple syntax:
1/2*((x-1)*(-4*x+1)^(1/2)-2*x^2-x+1)/x/(x-1)^2
Generating function in latex syntax:
\frac{\left(x -1\right) \sqrt{-4 x +1}-2 x^{2}-x +1}{2 x \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**2 - x/2 + sqrt(1 - 4*x)*(x - 1)/2 + 1/2)/(x*(x - 1)**2)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^4*F(x)^2+(x+1)*(2*x-1)*(x-1)^2*F(x)+x^3+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(x +1\right) \left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{3}+2 x^{2}-3 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+2) = -2*(3+2*n)/(3+n)*a(n)+(9+5*n)/(3+n)*a(1+n)+3*(1+n)/(3+n), n >= 4
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(n +2\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{3+n}+\frac{\left(9+5 n \right) a \! \left(1+n \right)}{3+n}+\frac{3+3 n}{3+n}, \quad n \geq 4
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21522/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[11,x]*F[3,x]
F[3,x] = F[1,x]+F[128,x]+F[4,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[7,x,k[0]]+F[9,x,k[0]]
F[7,x,k[0]] = F[6,x,k[0]]*F[8,x,k[0]]
F[8,x,k[0]] = k[0]*x
F[9,x,k[0]] = F[10,x,k[0]]*F[11,x]
F[10,x,k[0]] = (F[6,x,k[0]]*k[0]-F[6,x,1])/(-1+k[0])
F[11,x] = x
F[12,x,k[0]] = F[11,x]*F[13,x,k[0]]
F[13,x,k[0]] = F[14,x]+F[63,x,k[0]]
F[14,x] = F[15,x]+F[29,x]
F[15,x] = F[16,x]+F[26,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[11,x]*F[19,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[11,x]*F[20,x]
F[23,x] = F[21,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[11,x]*F[23,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[21,x]
F[29,x] = F[30,x]+F[62,x]
F[30,x] = F[31,x]+F[41,x]
F[31,x] = F[32,x]
F[32,x] = F[11,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[1,x]+F[11,x]
F[35,x] = F[31,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[11,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[11,x]
F[40,x] = F[36,x]
F[41,x] = F[42,x]+F[43,x]+F[52,x]
F[42,x] = 0
F[43,x] = F[11,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[17,x]+F[27,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[11,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[27,x]
F[51,x] = F[47,x]
F[52,x] = F[11,x]*F[53,x]
F[53,x] = F[23,x]+F[54,x]
F[54,x] = F[55,x]+F[60,x]
F[55,x] = F[42,x]+F[56,x]+F[58,x]
F[56,x] = F[11,x]*F[57,x]
F[57,x] = F[21,x]+F[55,x]
F[58,x] = F[11,x]*F[59,x]
F[59,x] = F[21,x]+F[55,x]
F[60,x] = F[61,x]
F[61,x] = F[11,x]*F[54,x]
F[62,x] = F[36,x]+F[47,x]
F[63,x,k[0]] = F[64,x,k[0]]+F[92,x,k[0]]
F[64,x,k[0]] = F[65,x,k[0]]+F[85,x,k[0]]
F[65,x,k[0]] = F[66,x,k[0]]+F[69,x,k[0]]
F[66,x,k[0]] = F[67,x,k[0]]
F[67,x,k[0]] = F[68,x,k[0]]*F[8,x,k[0]]
F[68,x,k[0]] = F[1,x]+F[66,x,k[0]]
F[69,x,k[0]] = F[42,x]+F[70,x,k[0]]+F[72,x,k[0]]
F[70,x,k[0]] = F[71,x,k[0]]*F[8,x,k[0]]
F[71,x,k[0]] = F[17,x]+F[69,x,k[0]]
F[72,x,k[0]] = F[11,x]*F[73,x,k[0]]
F[73,x,k[0]] = F[74,x,k[0]]+F[77,x,k[0]]
F[74,x,k[0]] = F[66,x,k[0]]+F[75,x,k[0]]
F[75,x,k[0]] = F[76,x,k[0]]
F[76,x,k[0]] = F[11,x]*F[74,x,k[0]]
F[77,x,k[0]] = F[78,x,k[0]]+F[83,x,k[0]]
F[78,x,k[0]] = F[42,x]+F[79,x,k[0]]+F[81,x,k[0]]
F[79,x,k[0]] = F[8,x,k[0]]*F[80,x,k[0]]
F[80,x,k[0]] = F[21,x]+F[78,x,k[0]]
F[81,x,k[0]] = F[11,x]*F[82,x,k[0]]
F[82,x,k[0]] = F[66,x,k[0]]+F[78,x,k[0]]
F[83,x,k[0]] = F[84,x,k[0]]
F[84,x,k[0]] = F[11,x]*F[77,x,k[0]]
F[85,x,k[0]] = F[86,x,k[0]]+F[89,x,k[0]]
F[86,x,k[0]] = F[87,x,k[0]]
F[87,x,k[0]] = F[8,x,k[0]]*F[88,x,k[0]]
F[88,x,k[0]] = F[11,x]+F[86,x,k[0]]
F[89,x,k[0]] = F[90,x,k[0]]
F[90,x,k[0]] = F[8,x,k[0]]*F[91,x,k[0]]
F[91,x,k[0]] = F[27,x]+F[89,x,k[0]]
F[92,x,k[0]] = F[127,x,k[0]]+F[93,x,k[0]]
F[93,x,k[0]] = F[104,x,k[0]]+F[94,x,k[0]]
F[94,x,k[0]] = F[42,x]+F[87,x,k[0]]+F[95,x,k[0]]
F[95,x,k[0]] = F[11,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[66,x,k[0]]+F[86,x,k[0]]
F[98,x,k[0]] = F[94,x,k[0]]+F[99,x,k[0]]
F[99,x,k[0]] = F[100,x,k[0]]
F[100,x,k[0]] = F[101,x,k[0]]*F[11,x]
F[101,x,k[0]] = F[102,x,k[0]]+F[103,x,k[0]]
F[102,x,k[0]] = F[86,x,k[0]]
F[103,x,k[0]] = F[99,x,k[0]]
F[104,x,k[0]] = F[105,x,k[0]]+F[114,x,k[0]]+F[42,x]+F[90,x,k[0]]
F[105,x,k[0]] = F[106,x,k[0]]*F[11,x]
F[106,x,k[0]] = F[107,x,k[0]]+F[108,x,k[0]]
F[107,x,k[0]] = F[69,x,k[0]]+F[89,x,k[0]]
F[108,x,k[0]] = F[104,x,k[0]]+F[109,x,k[0]]
F[109,x,k[0]] = F[110,x,k[0]]
F[110,x,k[0]] = F[11,x]*F[111,x,k[0]]
F[111,x,k[0]] = F[112,x,k[0]]+F[113,x,k[0]]
F[112,x,k[0]] = F[89,x,k[0]]
F[113,x,k[0]] = F[109,x,k[0]]
F[114,x,k[0]] = F[11,x]*F[115,x,k[0]]
F[115,x,k[0]] = F[116,x,k[0]]+F[119,x,k[0]]
F[116,x,k[0]] = F[117,x,k[0]]+F[75,x,k[0]]
F[117,x,k[0]] = F[118,x,k[0]]
F[118,x,k[0]] = F[11,x]*F[116,x,k[0]]
F[119,x,k[0]] = F[120,x,k[0]]+F[125,x,k[0]]
F[120,x,k[0]] = 2*F[42,x]+F[121,x,k[0]]+F[123,x,k[0]]
F[121,x,k[0]] = F[11,x]*F[122,x,k[0]]
F[122,x,k[0]] = F[120,x,k[0]]+F[78,x,k[0]]
F[123,x,k[0]] = F[11,x]*F[124,x,k[0]]
F[124,x,k[0]] = F[120,x,k[0]]+F[75,x,k[0]]
F[125,x,k[0]] = F[126,x,k[0]]
F[126,x,k[0]] = F[11,x]*F[119,x,k[0]]
F[127,x,k[0]] = F[109,x,k[0]]+F[99,x,k[0]]
F[128,x] = F[11,x]*F[14,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_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{128}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \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_{12}\! \left(x , y\right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = y x
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{10}\! \left(x , y\right) = \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}
F_{11}\! \left(x \right) = x
F_{12}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{14}\! \left(x \right)+F_{63}\! \left(x , y\right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \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_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{21}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{62}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{42}\! \left(x \right) = 0
F_{43}\! \left(x \right) = F_{11}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{27}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{55}\! \left(x \right) = F_{42}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{11}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{21}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{21}\! \left(x \right)+F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right) F_{54}\! \left(x \right)
F_{62}\! \left(x \right) = F_{36}\! \left(x \right)+F_{47}\! \left(x \right)
F_{63}\! \left(x , y\right) = F_{64}\! \left(x , y\right)+F_{92}\! \left(x , y\right)
F_{64}\! \left(x , y\right) = F_{65}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{66}\! \left(x , y\right)+F_{69}\! \left(x , y\right)
F_{66}\! \left(x , y\right) = F_{67}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{68}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{66}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = F_{42}\! \left(x \right)+F_{70}\! \left(x , y\right)+F_{72}\! \left(x , y\right)
F_{70}\! \left(x , y\right) = F_{71}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{71}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{69}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{73}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{74}\! \left(x , y\right)+F_{77}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{66}\! \left(x , y\right)+F_{75}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{76}\! \left(x , y\right)
F_{76}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{74}\! \left(x , y\right)
F_{77}\! \left(x , y\right) = F_{78}\! \left(x , y\right)+F_{83}\! \left(x , y\right)
F_{78}\! \left(x , y\right) = F_{42}\! \left(x \right)+F_{79}\! \left(x , y\right)+F_{81}\! \left(x , y\right)
F_{79}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{80}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{78}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{82}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{66}\! \left(x , y\right)+F_{78}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{77}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{86}\! \left(x , y\right)+F_{89}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{87}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{88}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{11}\! \left(x \right)+F_{86}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{91}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{27}\! \left(x \right)+F_{89}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{127}\! \left(x , y\right)+F_{93}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = F_{104}\! \left(x , y\right)+F_{94}\! \left(x , y\right)
F_{94}\! \left(x , y\right) = F_{42}\! \left(x \right)+F_{87}\! \left(x , y\right)+F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{11}\! \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_{66}\! \left(x , y\right)+F_{86}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{94}\! \left(x , y\right)+F_{99}\! \left(x , y\right)
F_{99}\! \left(x , y\right) = F_{100}\! \left(x , y\right)
F_{100}\! \left(x , y\right) = F_{101}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{101}\! \left(x , y\right) = F_{102}\! \left(x , y\right)+F_{103}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{86}\! \left(x , y\right)
F_{103}\! \left(x , y\right) = F_{99}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = F_{105}\! \left(x , y\right)+F_{114}\! \left(x , y\right)+F_{42}\! \left(x \right)+F_{90}\! \left(x , y\right)
F_{105}\! \left(x , y\right) = F_{106}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{106}\! \left(x , y\right) = F_{107}\! \left(x , y\right)+F_{108}\! \left(x , y\right)
F_{107}\! \left(x , y\right) = F_{69}\! \left(x , y\right)+F_{89}\! \left(x , y\right)
F_{108}\! \left(x , y\right) = F_{104}\! \left(x , y\right)+F_{109}\! \left(x , y\right)
F_{109}\! \left(x , y\right) = F_{110}\! \left(x , y\right)
F_{110}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{111}\! \left(x , y\right)
F_{111}\! \left(x , y\right) = F_{112}\! \left(x , y\right)+F_{113}\! \left(x , y\right)
F_{112}\! \left(x , y\right) = F_{89}\! \left(x , y\right)
F_{113}\! \left(x , y\right) = F_{109}\! \left(x , y\right)
F_{114}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{115}\! \left(x , y\right)
F_{115}\! \left(x , y\right) = F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)
F_{116}\! \left(x , y\right) = F_{117}\! \left(x , y\right)+F_{75}\! \left(x , y\right)
F_{117}\! \left(x , y\right) = F_{118}\! \left(x , y\right)
F_{118}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{116}\! \left(x , y\right)
F_{119}\! \left(x , y\right) = F_{120}\! \left(x , y\right)+F_{125}\! \left(x , y\right)
F_{120}\! \left(x , y\right) = 2 F_{42}\! \left(x \right)+F_{121}\! \left(x , y\right)+F_{123}\! \left(x , y\right)
F_{121}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{122}\! \left(x , y\right)
F_{122}\! \left(x , y\right) = F_{120}\! \left(x , y\right)+F_{78}\! \left(x , y\right)
F_{123}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{124}\! \left(x , y\right)
F_{124}\! \left(x , y\right) = F_{120}\! \left(x , y\right)+F_{75}\! \left(x , y\right)
F_{125}\! \left(x , y\right) = F_{126}\! \left(x , y\right)
F_{126}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{119}\! \left(x , y\right)
F_{127}\! \left(x , y\right) = F_{109}\! \left(x , y\right)+F_{99}\! \left(x , y\right)
F_{128}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \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_11(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_128(x) + F_4(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_12(x, k_0) + F_7(x, k_0) + F_9(x, k_0))
Eq(F_7(x, k_0), F_6(x, k_0)*F_8(x, k_0))
Eq(F_8(x, k_0), k_0*x)
Eq(F_9(x, k_0), F_10(x, k_0)*F_11(x))
Eq(F_10(x, k_0), (-k_0*F_6(x, k_0) + F_6(x, 1))/(1 - k_0))
Eq(F_11(x), x)
Eq(F_12(x, k_0), F_11(x)*F_13(x, k_0))
Eq(F_13(x, k_0), F_14(x) + F_63(x, k_0))
Eq(F_14(x), F_15(x) + F_29(x))
Eq(F_15(x), F_16(x) + F_26(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_11(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_20(x))
Eq(F_23(x), F_21(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_11(x)*F_23(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_21(x))
Eq(F_29(x), F_30(x) + F_62(x))
Eq(F_30(x), F_31(x) + F_41(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_11(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_1(x) + F_11(x))
Eq(F_35(x), F_31(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_11(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_11(x))
Eq(F_40(x), F_36(x))
Eq(F_41(x), F_42(x) + F_43(x) + F_52(x))
Eq(F_42(x), 0)
Eq(F_43(x), F_11(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_17(x) + F_27(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_11(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_27(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_11(x)*F_53(x))
Eq(F_53(x), F_23(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_60(x))
Eq(F_55(x), F_42(x) + F_56(x) + F_58(x))
Eq(F_56(x), F_11(x)*F_57(x))
Eq(F_57(x), F_21(x) + F_55(x))
Eq(F_58(x), F_11(x)*F_59(x))
Eq(F_59(x), F_21(x) + F_55(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_11(x)*F_54(x))
Eq(F_62(x), F_36(x) + F_47(x))
Eq(F_63(x, k_0), F_64(x, k_0) + F_92(x, k_0))
Eq(F_64(x, k_0), F_65(x, k_0) + F_85(x, k_0))
Eq(F_65(x, k_0), F_66(x, k_0) + F_69(x, k_0))
Eq(F_66(x, k_0), F_67(x, k_0))
Eq(F_67(x, k_0), F_68(x, k_0)*F_8(x, k_0))
Eq(F_68(x, k_0), F_1(x) + F_66(x, k_0))
Eq(F_69(x, k_0), F_42(x) + F_70(x, k_0) + F_72(x, k_0))
Eq(F_70(x, k_0), F_71(x, k_0)*F_8(x, k_0))
Eq(F_71(x, k_0), F_17(x) + F_69(x, k_0))
Eq(F_72(x, k_0), F_11(x)*F_73(x, k_0))
Eq(F_73(x, k_0), F_74(x, k_0) + F_77(x, k_0))
Eq(F_74(x, k_0), F_66(x, k_0) + F_75(x, k_0))
Eq(F_75(x, k_0), F_76(x, k_0))
Eq(F_76(x, k_0), F_11(x)*F_74(x, k_0))
Eq(F_77(x, k_0), F_78(x, k_0) + F_83(x, k_0))
Eq(F_78(x, k_0), F_42(x) + F_79(x, k_0) + F_81(x, k_0))
Eq(F_79(x, k_0), F_8(x, k_0)*F_80(x, k_0))
Eq(F_80(x, k_0), F_21(x) + F_78(x, k_0))
Eq(F_81(x, k_0), F_11(x)*F_82(x, k_0))
Eq(F_82(x, k_0), F_66(x, k_0) + F_78(x, k_0))
Eq(F_83(x, k_0), F_84(x, k_0))
Eq(F_84(x, k_0), F_11(x)*F_77(x, k_0))
Eq(F_85(x, k_0), F_86(x, k_0) + F_89(x, k_0))
Eq(F_86(x, k_0), F_87(x, k_0))
Eq(F_87(x, k_0), F_8(x, k_0)*F_88(x, k_0))
Eq(F_88(x, k_0), F_11(x) + F_86(x, k_0))
Eq(F_89(x, k_0), F_90(x, k_0))
Eq(F_90(x, k_0), F_8(x, k_0)*F_91(x, k_0))
Eq(F_91(x, k_0), F_27(x) + F_89(x, k_0))
Eq(F_92(x, k_0), F_127(x, k_0) + F_93(x, k_0))
Eq(F_93(x, k_0), F_104(x, k_0) + F_94(x, k_0))
Eq(F_94(x, k_0), F_42(x) + F_87(x, k_0) + F_95(x, k_0))
Eq(F_95(x, k_0), F_11(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_66(x, k_0) + F_86(x, k_0))
Eq(F_98(x, k_0), F_94(x, k_0) + F_99(x, k_0))
Eq(F_99(x, k_0), F_100(x, k_0))
Eq(F_100(x, k_0), F_101(x, k_0)*F_11(x))
Eq(F_101(x, k_0), F_102(x, k_0) + F_103(x, k_0))
Eq(F_102(x, k_0), F_86(x, k_0))
Eq(F_103(x, k_0), F_99(x, k_0))
Eq(F_104(x, k_0), F_105(x, k_0) + F_114(x, k_0) + F_42(x) + F_90(x, k_0))
Eq(F_105(x, k_0), F_106(x, k_0)*F_11(x))
Eq(F_106(x, k_0), F_107(x, k_0) + F_108(x, k_0))
Eq(F_107(x, k_0), F_69(x, k_0) + F_89(x, k_0))
Eq(F_108(x, k_0), F_104(x, k_0) + F_109(x, k_0))
Eq(F_109(x, k_0), F_110(x, k_0))
Eq(F_110(x, k_0), F_11(x)*F_111(x, k_0))
Eq(F_111(x, k_0), F_112(x, k_0) + F_113(x, k_0))
Eq(F_112(x, k_0), F_89(x, k_0))
Eq(F_113(x, k_0), F_109(x, k_0))
Eq(F_114(x, k_0), F_11(x)*F_115(x, k_0))
Eq(F_115(x, k_0), F_116(x, k_0) + F_119(x, k_0))
Eq(F_116(x, k_0), F_117(x, k_0) + F_75(x, k_0))
Eq(F_117(x, k_0), F_118(x, k_0))
Eq(F_118(x, k_0), F_11(x)*F_116(x, k_0))
Eq(F_119(x, k_0), F_120(x, k_0) + F_125(x, k_0))
Eq(F_120(x, k_0), F_121(x, k_0) + F_123(x, k_0) + 2*F_42(x))
Eq(F_121(x, k_0), F_11(x)*F_122(x, k_0))
Eq(F_122(x, k_0), F_120(x, k_0) + F_78(x, k_0))
Eq(F_123(x, k_0), F_11(x)*F_124(x, k_0))
Eq(F_124(x, k_0), F_120(x, k_0) + F_75(x, k_0))
Eq(F_125(x, k_0), F_126(x, k_0))
Eq(F_126(x, k_0), F_11(x)*F_119(x, k_0))
Eq(F_127(x, k_0), F_109(x, k_0) + F_99(x, k_0))
Eq(F_128(x), F_11(x)*F_14(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, 2, 3], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 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": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 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], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 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, 2, 3], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], 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