0123_0231_1203_1230
Counting sequence:
1, 1, 2, 6, 20, 68, 233, 805, 2807, 9879, 35073, 125513, 452389, 1641029, 5986994, 21954974, 80884424, 299233544, 1111219334, 4140813374, 15478839554, 58028869154, 218123355524, 821908275548, 3104046382352, 11747506651600, 44546351423300, 169227201341652, 643973514492300, 2454475583281868, 9369138618324169, 35813931416918549, 137081932075065263, 525353713400309807, 2015751124008149541, 7742978520054000509, 29774186072954835889, 114606440210718052561, 441564086366680450151, 1702840385183220958191, 6572504892268134874571, 25389097620896782565771, 98153717002173246307991, 379746878184999236074631, 1470264493076253923152271, 5696357496454931166691151, 22084473830579631273028001, 85674824558366902978160921, 332571273253645310076785693, 1291728440650571687591343005, 5019990795652566765574610249, 19519586400396559945255395449, 75939123964633742295524406305, 295583099081022345242026223969, 1151078279633934273339287081093, 4484705974261170028903985793461, 17480744975346274938049889777501, 68167299550918875163057766764253, 265935374100251252826704572507205, 1037896348441217324052190464811877, 4052322377833832889165725753950421, 15827804203709748413903022605791061, 61844179276390474550415490459624562, 241731648034663174453553763280411262, 945191881433949524368025572097182712, 3697036319692422921616961489458597112, 14465398172528314914987682691817256862, 56616872127575565714456218127404465462, 221664526639870969818868730255060873162, 868115630039721289195243320535173168362, 3400841337845262307634299160149779795512, 13326591878262451051725417340414289388512, 52236300463921029167075657836254276221612, 204806200044869594622195474090364872479852, 803208097853041764175883613060708735216392, 3150825719355588051846748377439648934658872, 12363150585438864804401337239737465205859872, 48522108743351554113668766536209704058695392, 190481702695551536648572308856405812844416422, 747941118292358009671637493802091734983245902, 2937512012386005948271613037298890005147672762, 11539516928026141818080270960202279274544597562, 45340788029572782856498041959039615144737335102, 178189356803407511469364401178870887279438203342, 700428157646891111385116132644010213161199046982, 2753800825358868173953738644727964754143743552262, 10828955512911245501993491949149035980653831318562, 42591586429343306278351839513769445156600185789682, 167549223561818274061312005320318729317784487294282, 659235507754171462942804374299177299507150833191882, 2594278539737466584501859445989443080556260333314082, 10211011895061346620794479264241087293022128101102082, 40197172785417983558000388917514049720963994752868882, 158268765232814157223991038379331474943305364426222482, 623255872174088354738765355587317625767974467684667332, 2454751480625186324597424156056685091803019648593553532, 9669794427942852990486560919955224534096961204735920032, 38097293785285059898640505834684015344691782927693020832, 150119408440311693996084645014287156632692002279720846172, 591621693941984491359615634297010081141446158615384660812, 2331925120646357396693534005617323073487777818072950249492
Generating function in Maple syntax:
-1/2*(-1+2*x+(-4*x+1)^(1/2))/x^2*(-1+2*x)/(-1+x)
Generating function in latex syntax:
-\frac{\left(-1+2 x +\sqrt{-4 x +1}\right) \left(-1+2 x \right)}{2 x^{2} \left(-1+x \right)}
Generating function in sympy syntax:
(2*x - 1)*(-x - sqrt(1 - 4*x)/2 + 1/2)/(x**2*(x - 1))
Implicit equation for the generating function in Maple syntax:
x^2*(-1+x)^2*F(x)^2+(-1+x)*(-1+2*x)^2*F(x)+(-1+2*x)^2 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(-1+x \right)^{2} F \! \left(x \right)^{2}+\left(-1+x \right) \left(-1+2 x \right)^{2} F \! \left(x \right)+\left(-1+2 x \right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+3) = 4*(3+2*n)/(5+n)*a(n)+(25+7*n)/(5+n)*a(2+n)-2*(16+7*n)/(5+n)*a(n+1), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +3\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(25+7 n \right) a \! \left(2+n \right)}{5+n}-\frac{2 \left(16+7 n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21152/
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[136,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[64,x,k[0]]
F[14,x] = F[15,x]+F[36,x]
F[15,x] = F[16,x]+F[31,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[26,x]+F[28,x]
F[25,x] = 0
F[26,x] = F[11,x]*F[27,x]
F[27,x] = F[21,x]+F[24,x]
F[28,x] = F[11,x]*F[29,x]
F[29,x] = F[21,x]+F[30,x]
F[30,x] = F[28,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]*F[34,x]
F[34,x] = F[21,x]+F[35,x]
F[35,x] = F[33,x]
F[36,x] = F[37,x]+F[57,x]
F[37,x] = F[21,x]+F[38,x]
F[38,x] = F[25,x]+F[39,x]+F[41,x]
F[39,x] = F[11,x]*F[40,x]
F[40,x] = F[17,x]+F[38,x]
F[41,x] = F[11,x]*F[42,x]
F[42,x] = F[43,x]+F[48,x]
F[43,x] = F[21,x]+F[44,x]
F[44,x] = F[25,x]+F[45,x]+F[47,x]
F[45,x] = F[11,x]*F[46,x]
F[46,x] = F[21,x]+F[44,x]
F[47,x] = F[11,x]*F[43,x]
F[48,x] = F[44,x]+F[49,x]
F[49,x] = F[25,x]+F[50,x]+F[53,x]+F[55,x]
F[50,x] = F[11,x]*F[51,x]
F[51,x] = F[30,x]+F[52,x]
F[52,x] = 2*F[25,x]+F[50,x]+F[53,x]
F[53,x] = F[11,x]*F[54,x]
F[54,x] = F[44,x]+F[52,x]
F[55,x] = F[11,x]*F[56,x]
F[56,x] = F[44,x]+F[49,x]
F[57,x] = F[24,x]+F[58,x]
F[58,x] = 2*F[25,x]+F[59,x]+F[61,x]
F[59,x] = F[11,x]*F[60,x]
F[60,x] = F[32,x]+F[58,x]
F[61,x] = F[11,x]*F[62,x]
F[62,x] = F[44,x]+F[63,x]
F[63,x] = F[61,x]
F[64,x,k[0]] = F[102,x,k[0]]+F[65,x,k[0]]
F[65,x,k[0]] = F[66,x,k[0]]+F[89,x,k[0]]
F[66,x,k[0]] = F[67,x,k[0]]+F[70,x,k[0]]
F[67,x,k[0]] = F[68,x,k[0]]
F[68,x,k[0]] = F[69,x,k[0]]*F[8,x,k[0]]
F[69,x,k[0]] = F[1,x]+F[67,x,k[0]]
F[70,x,k[0]] = F[25,x]+F[71,x,k[0]]+F[73,x,k[0]]
F[71,x,k[0]] = F[72,x,k[0]]*F[8,x,k[0]]
F[72,x,k[0]] = F[17,x]+F[70,x,k[0]]
F[73,x,k[0]] = F[11,x]*F[74,x,k[0]]
F[74,x,k[0]] = F[75,x,k[0]]+F[80,x,k[0]]
F[75,x,k[0]] = F[67,x,k[0]]+F[76,x,k[0]]
F[76,x,k[0]] = F[25,x]+F[77,x,k[0]]+F[79,x,k[0]]
F[77,x,k[0]] = F[78,x,k[0]]*F[8,x,k[0]]
F[78,x,k[0]] = F[21,x]+F[76,x,k[0]]
F[79,x,k[0]] = F[11,x]*F[75,x,k[0]]
F[80,x,k[0]] = F[76,x,k[0]]+F[81,x,k[0]]
F[81,x,k[0]] = F[25,x]+F[82,x,k[0]]+F[85,x,k[0]]+F[87,x,k[0]]
F[82,x,k[0]] = F[8,x,k[0]]*F[83,x,k[0]]
F[83,x,k[0]] = F[30,x]+F[84,x,k[0]]
F[84,x,k[0]] = 2*F[25,x]+F[82,x,k[0]]+F[85,x,k[0]]
F[85,x,k[0]] = F[11,x]*F[86,x,k[0]]
F[86,x,k[0]] = F[76,x,k[0]]+F[84,x,k[0]]
F[87,x,k[0]] = F[11,x]*F[88,x,k[0]]
F[88,x,k[0]] = F[76,x,k[0]]+F[81,x,k[0]]
F[89,x,k[0]] = F[90,x,k[0]]+F[96,x,k[0]]
F[90,x,k[0]] = F[25,x]+F[91,x,k[0]]+F[93,x,k[0]]
F[91,x,k[0]] = F[8,x,k[0]]*F[92,x,k[0]]
F[92,x,k[0]] = F[21,x]+F[90,x,k[0]]
F[93,x,k[0]] = F[11,x]*F[94,x,k[0]]
F[94,x,k[0]] = F[67,x,k[0]]+F[95,x,k[0]]
F[95,x,k[0]] = F[93,x,k[0]]
F[96,x,k[0]] = 2*F[25,x]+F[97,x,k[0]]+F[99,x,k[0]]
F[97,x,k[0]] = F[8,x,k[0]]*F[98,x,k[0]]
F[98,x,k[0]] = F[32,x]+F[96,x,k[0]]
F[99,x,k[0]] = F[100,x,k[0]]*F[11,x]
F[100,x,k[0]] = F[101,x,k[0]]+F[76,x,k[0]]
F[101,x,k[0]] = F[99,x,k[0]]
F[102,x,k[0]] = F[103,x,k[0]]+F[123,x,k[0]]
F[103,x,k[0]] = F[104,x,k[0]]+F[95,x,k[0]]
F[104,x,k[0]] = 2*F[25,x]+F[105,x,k[0]]+F[107,x,k[0]]
F[105,x,k[0]] = F[106,x,k[0]]*F[11,x]
F[106,x,k[0]] = F[104,x,k[0]]+F[70,x,k[0]]
F[107,x,k[0]] = F[108,x,k[0]]*F[11,x]
F[108,x,k[0]] = F[109,x,k[0]]+F[114,x,k[0]]
F[109,x,k[0]] = F[110,x,k[0]]+F[95,x,k[0]]
F[110,x,k[0]] = 2*F[25,x]+F[111,x,k[0]]+F[113,x,k[0]]
F[111,x,k[0]] = F[11,x]*F[112,x,k[0]]
F[112,x,k[0]] = F[110,x,k[0]]+F[76,x,k[0]]
F[113,x,k[0]] = F[109,x,k[0]]*F[11,x]
F[114,x,k[0]] = F[110,x,k[0]]+F[115,x,k[0]]
F[115,x,k[0]] = 2*F[25,x]+F[116,x,k[0]]+F[119,x,k[0]]+F[121,x,k[0]]
F[116,x,k[0]] = F[11,x]*F[117,x,k[0]]
F[117,x,k[0]] = F[118,x,k[0]]+F[84,x,k[0]]
F[118,x,k[0]] = 3*F[25,x]+F[116,x,k[0]]+F[119,x,k[0]]
F[119,x,k[0]] = F[11,x]*F[120,x,k[0]]
F[120,x,k[0]] = F[110,x,k[0]]+F[118,x,k[0]]
F[121,x,k[0]] = F[11,x]*F[122,x,k[0]]
F[122,x,k[0]] = F[110,x,k[0]]+F[115,x,k[0]]
F[123,x,k[0]] = F[124,x,k[0]]+F[130,x,k[0]]
F[124,x,k[0]] = 2*F[25,x]+F[125,x,k[0]]+F[127,x,k[0]]
F[125,x,k[0]] = F[11,x]*F[126,x,k[0]]
F[126,x,k[0]] = F[124,x,k[0]]+F[90,x,k[0]]
F[127,x,k[0]] = F[11,x]*F[128,x,k[0]]
F[128,x,k[0]] = F[129,x,k[0]]+F[95,x,k[0]]
F[129,x,k[0]] = F[127,x,k[0]]
F[130,x,k[0]] = 3*F[25,x]+F[131,x,k[0]]+F[133,x,k[0]]
F[131,x,k[0]] = F[11,x]*F[132,x,k[0]]
F[132,x,k[0]] = F[130,x,k[0]]+F[96,x,k[0]]
F[133,x,k[0]] = F[11,x]*F[134,x,k[0]]
F[134,x,k[0]] = F[110,x,k[0]]+F[135,x,k[0]]
F[135,x,k[0]] = F[133,x,k[0]]
F[136,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_{136}\! \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_{64}\! \left(x , y\right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{31}\! \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_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{11}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{21}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right) F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{25}\! \left(x \right)+F_{50}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{30}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{57}\! \left(x \right) = F_{24}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{32}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{44}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{61}\! \left(x \right)
F_{64}\! \left(x , y\right) = F_{102}\! \left(x , y\right)+F_{65}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{66}\! \left(x , y\right)+F_{89}\! \left(x , y\right)
F_{66}\! \left(x , y\right) = F_{67}\! \left(x , y\right)+F_{70}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{68}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = F_{69}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{67}\! \left(x , y\right)
F_{70}\! \left(x , y\right) = F_{25}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)
F_{71}\! \left(x , y\right) = F_{72}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{70}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{74}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{75}\! \left(x , y\right)+F_{80}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{67}\! \left(x , y\right)+F_{76}\! \left(x , y\right)
F_{76}\! \left(x , y\right) = F_{25}\! \left(x \right)+F_{77}\! \left(x , y\right)+F_{79}\! \left(x , y\right)
F_{77}\! \left(x , y\right) = F_{78}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{78}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{76}\! \left(x , y\right)
F_{79}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{75}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{76}\! \left(x , y\right)+F_{81}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{25}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{85}\! \left(x , y\right)+F_{87}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{83}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{30}\! \left(x \right)+F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = 2 F_{25}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{86}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{76}\! \left(x , y\right)+F_{84}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{88}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{76}\! \left(x , y\right)+F_{81}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{90}\! \left(x , y\right)+F_{96}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{25}\! \left(x \right)+F_{91}\! \left(x , y\right)+F_{93}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{92}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{90}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{94}\! \left(x , y\right)
F_{94}\! \left(x , y\right) = F_{67}\! \left(x , y\right)+F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{93}\! \left(x , y\right)
F_{96}\! \left(x , y\right) = 2 F_{25}\! \left(x \right)+F_{97}\! \left(x , y\right)+F_{99}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{98}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{32}\! \left(x \right)+F_{96}\! \left(x , y\right)
F_{99}\! \left(x , y\right) = F_{100}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{100}\! \left(x , y\right) = F_{101}\! \left(x , y\right)+F_{76}\! \left(x , y\right)
F_{101}\! \left(x , y\right) = F_{99}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{103}\! \left(x , y\right)+F_{123}\! \left(x , y\right)
F_{103}\! \left(x , y\right) = F_{104}\! \left(x , y\right)+F_{95}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = 2 F_{25}\! \left(x \right)+F_{105}\! \left(x , y\right)+F_{107}\! \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_{104}\! \left(x , y\right)+F_{70}\! \left(x , y\right)
F_{107}\! \left(x , y\right) = F_{108}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{108}\! \left(x , y\right) = F_{109}\! \left(x , y\right)+F_{114}\! \left(x , y\right)
F_{109}\! \left(x , y\right) = F_{110}\! \left(x , y\right)+F_{95}\! \left(x , y\right)
F_{110}\! \left(x , y\right) = 2 F_{25}\! \left(x \right)+F_{111}\! \left(x , y\right)+F_{113}\! \left(x , y\right)
F_{111}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{112}\! \left(x , y\right)
F_{112}\! \left(x , y\right) = F_{110}\! \left(x , y\right)+F_{76}\! \left(x , y\right)
F_{113}\! \left(x , y\right) = F_{109}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{114}\! \left(x , y\right) = F_{110}\! \left(x , y\right)+F_{115}\! \left(x , y\right)
F_{115}\! \left(x , y\right) = 2 F_{25}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)
F_{116}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{117}\! \left(x , y\right)
F_{117}\! \left(x , y\right) = F_{118}\! \left(x , y\right)+F_{84}\! \left(x , y\right)
F_{118}\! \left(x , y\right) = 3 F_{25}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)
F_{119}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{120}\! \left(x , y\right)
F_{120}\! \left(x , y\right) = F_{110}\! \left(x , y\right)+F_{118}\! \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_{110}\! \left(x , y\right)+F_{115}\! \left(x , y\right)
F_{123}\! \left(x , y\right) = F_{124}\! \left(x , y\right)+F_{130}\! \left(x , y\right)
F_{124}\! \left(x , y\right) = 2 F_{25}\! \left(x \right)+F_{125}\! \left(x , y\right)+F_{127}\! \left(x , y\right)
F_{125}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{126}\! \left(x , y\right)
F_{126}\! \left(x , y\right) = F_{124}\! \left(x , y\right)+F_{90}\! \left(x , y\right)
F_{127}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{128}\! \left(x , y\right)
F_{128}\! \left(x , y\right) = F_{129}\! \left(x , y\right)+F_{95}\! \left(x , y\right)
F_{129}\! \left(x , y\right) = F_{127}\! \left(x , y\right)
F_{130}\! \left(x , y\right) = 3 F_{25}\! \left(x \right)+F_{131}\! \left(x , y\right)+F_{133}\! \left(x , y\right)
F_{131}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{132}\! \left(x , y\right)
F_{132}\! \left(x , y\right) = F_{130}\! \left(x , y\right)+F_{96}\! \left(x , y\right)
F_{133}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{134}\! \left(x , y\right)
F_{134}\! \left(x , y\right) = F_{110}\! \left(x , y\right)+F_{135}\! \left(x , y\right)
F_{135}\! \left(x , y\right) = F_{133}\! \left(x , y\right)
F_{136}\! \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_136(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_64(x, k_0))
Eq(F_14(x), F_15(x) + F_36(x))
Eq(F_15(x), F_16(x) + F_31(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) + F_26(x) + F_28(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_11(x)*F_27(x))
Eq(F_27(x), F_21(x) + F_24(x))
Eq(F_28(x), F_11(x)*F_29(x))
Eq(F_29(x), F_21(x) + F_30(x))
Eq(F_30(x), F_28(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x)*F_34(x))
Eq(F_34(x), F_21(x) + F_35(x))
Eq(F_35(x), F_33(x))
Eq(F_36(x), F_37(x) + F_57(x))
Eq(F_37(x), F_21(x) + F_38(x))
Eq(F_38(x), F_25(x) + F_39(x) + F_41(x))
Eq(F_39(x), F_11(x)*F_40(x))
Eq(F_40(x), F_17(x) + F_38(x))
Eq(F_41(x), F_11(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_48(x))
Eq(F_43(x), F_21(x) + F_44(x))
Eq(F_44(x), F_25(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_11(x)*F_46(x))
Eq(F_46(x), F_21(x) + F_44(x))
Eq(F_47(x), F_11(x)*F_43(x))
Eq(F_48(x), F_44(x) + F_49(x))
Eq(F_49(x), F_25(x) + F_50(x) + F_53(x) + F_55(x))
Eq(F_50(x), F_11(x)*F_51(x))
Eq(F_51(x), F_30(x) + F_52(x))
Eq(F_52(x), 2*F_25(x) + F_50(x) + F_53(x))
Eq(F_53(x), F_11(x)*F_54(x))
Eq(F_54(x), F_44(x) + F_52(x))
Eq(F_55(x), F_11(x)*F_56(x))
Eq(F_56(x), F_44(x) + F_49(x))
Eq(F_57(x), F_24(x) + F_58(x))
Eq(F_58(x), 2*F_25(x) + F_59(x) + F_61(x))
Eq(F_59(x), F_11(x)*F_60(x))
Eq(F_60(x), F_32(x) + F_58(x))
Eq(F_61(x), F_11(x)*F_62(x))
Eq(F_62(x), F_44(x) + F_63(x))
Eq(F_63(x), F_61(x))
Eq(F_64(x, k_0), F_102(x, k_0) + F_65(x, k_0))
Eq(F_65(x, k_0), F_66(x, k_0) + F_89(x, k_0))
Eq(F_66(x, k_0), F_67(x, k_0) + F_70(x, k_0))
Eq(F_67(x, k_0), F_68(x, k_0))
Eq(F_68(x, k_0), F_69(x, k_0)*F_8(x, k_0))
Eq(F_69(x, k_0), F_1(x) + F_67(x, k_0))
Eq(F_70(x, k_0), F_25(x) + F_71(x, k_0) + F_73(x, k_0))
Eq(F_71(x, k_0), F_72(x, k_0)*F_8(x, k_0))
Eq(F_72(x, k_0), F_17(x) + F_70(x, k_0))
Eq(F_73(x, k_0), F_11(x)*F_74(x, k_0))
Eq(F_74(x, k_0), F_75(x, k_0) + F_80(x, k_0))
Eq(F_75(x, k_0), F_67(x, k_0) + F_76(x, k_0))
Eq(F_76(x, k_0), F_25(x) + F_77(x, k_0) + F_79(x, k_0))
Eq(F_77(x, k_0), F_78(x, k_0)*F_8(x, k_0))
Eq(F_78(x, k_0), F_21(x) + F_76(x, k_0))
Eq(F_79(x, k_0), F_11(x)*F_75(x, k_0))
Eq(F_80(x, k_0), F_76(x, k_0) + F_81(x, k_0))
Eq(F_81(x, k_0), F_25(x) + F_82(x, k_0) + F_85(x, k_0) + F_87(x, k_0))
Eq(F_82(x, k_0), F_8(x, k_0)*F_83(x, k_0))
Eq(F_83(x, k_0), F_30(x) + F_84(x, k_0))
Eq(F_84(x, k_0), 2*F_25(x) + F_82(x, k_0) + F_85(x, k_0))
Eq(F_85(x, k_0), F_11(x)*F_86(x, k_0))
Eq(F_86(x, k_0), F_76(x, k_0) + F_84(x, k_0))
Eq(F_87(x, k_0), F_11(x)*F_88(x, k_0))
Eq(F_88(x, k_0), F_76(x, k_0) + F_81(x, k_0))
Eq(F_89(x, k_0), F_90(x, k_0) + F_96(x, k_0))
Eq(F_90(x, k_0), F_25(x) + F_91(x, k_0) + F_93(x, k_0))
Eq(F_91(x, k_0), F_8(x, k_0)*F_92(x, k_0))
Eq(F_92(x, k_0), F_21(x) + F_90(x, k_0))
Eq(F_93(x, k_0), F_11(x)*F_94(x, k_0))
Eq(F_94(x, k_0), F_67(x, k_0) + F_95(x, k_0))
Eq(F_95(x, k_0), F_93(x, k_0))
Eq(F_96(x, k_0), 2*F_25(x) + F_97(x, k_0) + F_99(x, k_0))
Eq(F_97(x, k_0), F_8(x, k_0)*F_98(x, k_0))
Eq(F_98(x, k_0), F_32(x) + F_96(x, k_0))
Eq(F_99(x, k_0), F_100(x, k_0)*F_11(x))
Eq(F_100(x, k_0), F_101(x, k_0) + F_76(x, k_0))
Eq(F_101(x, k_0), F_99(x, k_0))
Eq(F_102(x, k_0), F_103(x, k_0) + F_123(x, k_0))
Eq(F_103(x, k_0), F_104(x, k_0) + F_95(x, k_0))
Eq(F_104(x, k_0), F_105(x, k_0) + F_107(x, k_0) + 2*F_25(x))
Eq(F_105(x, k_0), F_106(x, k_0)*F_11(x))
Eq(F_106(x, k_0), F_104(x, k_0) + F_70(x, k_0))
Eq(F_107(x, k_0), F_108(x, k_0)*F_11(x))
Eq(F_108(x, k_0), F_109(x, k_0) + F_114(x, k_0))
Eq(F_109(x, k_0), F_110(x, k_0) + F_95(x, k_0))
Eq(F_110(x, k_0), F_111(x, k_0) + F_113(x, k_0) + 2*F_25(x))
Eq(F_111(x, k_0), F_11(x)*F_112(x, k_0))
Eq(F_112(x, k_0), F_110(x, k_0) + F_76(x, k_0))
Eq(F_113(x, k_0), F_109(x, k_0)*F_11(x))
Eq(F_114(x, k_0), F_110(x, k_0) + F_115(x, k_0))
Eq(F_115(x, k_0), F_116(x, k_0) + F_119(x, k_0) + F_121(x, k_0) + 2*F_25(x))
Eq(F_116(x, k_0), F_11(x)*F_117(x, k_0))
Eq(F_117(x, k_0), F_118(x, k_0) + F_84(x, k_0))
Eq(F_118(x, k_0), F_116(x, k_0) + F_119(x, k_0) + 3*F_25(x))
Eq(F_119(x, k_0), F_11(x)*F_120(x, k_0))
Eq(F_120(x, k_0), F_110(x, k_0) + F_118(x, k_0))
Eq(F_121(x, k_0), F_11(x)*F_122(x, k_0))
Eq(F_122(x, k_0), F_110(x, k_0) + F_115(x, k_0))
Eq(F_123(x, k_0), F_124(x, k_0) + F_130(x, k_0))
Eq(F_124(x, k_0), F_125(x, k_0) + F_127(x, k_0) + 2*F_25(x))
Eq(F_125(x, k_0), F_11(x)*F_126(x, k_0))
Eq(F_126(x, k_0), F_124(x, k_0) + F_90(x, k_0))
Eq(F_127(x, k_0), F_11(x)*F_128(x, k_0))
Eq(F_128(x, k_0), F_129(x, k_0) + F_95(x, k_0))
Eq(F_129(x, k_0), F_127(x, k_0))
Eq(F_130(x, k_0), F_131(x, k_0) + F_133(x, k_0) + 3*F_25(x))
Eq(F_131(x, k_0), F_11(x)*F_132(x, k_0))
Eq(F_132(x, k_0), F_130(x, k_0) + F_96(x, k_0))
Eq(F_133(x, k_0), F_11(x)*F_134(x, k_0))
Eq(F_134(x, k_0), F_110(x, k_0) + F_135(x, k_0))
Eq(F_135(x, k_0), F_133(x, k_0))
Eq(F_136(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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], "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": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, 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