0123_0132_0213_0231_1023_1230
Counting sequence:
1, 1, 2, 6, 18, 56, 178, 579, 1923, 6506, 22365, 77933, 274718, 977979, 3511113, 12698137, 46218040, 169171277, 622312677, 2299457564, 8530613261, 31761811465, 118646747526, 444535638601, 1670121406171, 6290484218924, 23748150402599, 89848292734623, 340609688743278, 1293633096297261, 4921733383632123, 18755518085853557, 71581343342349270, 273583719761992119, 1047040975499975757, 4012237929344782918, 15393189206405750639, 59123660470557673397, 227331115692249218452, 874978539243539207511, 3370968971638267131983, 12999015359059752872734, 50170160158049479746317, 193794817538918788011711, 749176237466156072927588, 2898381950672152230871619, 11221286660485488307669447, 43474206660194191278463116, 168542734139782698284031823, 653831638465211933844244531, 2537980233876816554796067010, 9857493052497110597761910553, 38308116916634231795452846163, 148953910083885143873306970284, 579483411136927595216084490847, 2255536194918066741580763645343, 8783557097843960891451287581654, 34221216916319241795292312387717, 133388320447595759725142601852827, 520152626777942283442587915782328, 2029214991243413477787506659049891, 7919605586351057440255888535314331, 30920856391447857658381308182559742, 120772056619401183585598419969821526, 471892607400574447071565853773381594, 1844491536053351339953783398094308220, 7212107388262605203061246872289741014, 28209486799970634668628603095953268484, 110375155018072900127887659220245827798, 432002694284346475426582950543143821082, 1691362109724781074592041505626216976480, 6623963723634801158323997063254309264512, 25949372006228417554865152782754708177492, 101685685245648385435559507090368154893804, 398578154783561324402102954255422715047136, 1562732515729433765984006708089054462962960, 6128725844254092359439423287701723370980136, 24041809777932653076977269830862398765611096, 94334961372992596073450072556421722453877792, 370240566872570379454288465858599798097381038, 1453444618969834330539991111710394318740201570, 5707075073480195393548953183016145765808452888, 22414385552172134014960819058632860807609660858, 88051321658455883151471341681305387349404010366, 345969988218789581525604342993275503304892483104, 1359666130533958911234336238648720055091429485790, 5344601290230482991217351483234565883983730722334, 21012874849585321027574759366559219164996543794024, 82630576578465668920259356679598151594804307148718, 324996980062720932877649266903473245559613042179462, 1278495636270015554986314599147320898591515368756980, 5030356533308739580233910557256460359752065346907042, 19795940790469934978489753619109040761632589663598022, 77916549391951080939350520611668450993806320557895464, 306731120055514242272499396926697528297691875454051886, 1207699714354212699561530660831646278344789496431975400, 4755887347779527317060069045890637020339426414288262386, 18731568822308459037933265812469436866722709345150184786, 73787927830877435366053294383644436901663853989437669572, 290712468735156093191772344898488662201002336034329391498, 1145533627516962309680620308140148018855462248312999748390
Generating function in Maple syntax:
1/2*(-2*(-4*x+1)^(1/2)*x^2+4*x^2+(-4*x+1)^(1/2)-1)/(x^2+x-1)/x
Generating function in latex syntax:
\frac{-2 \sqrt{-4 x +1}\, x^{2}+4 x^{2}+\sqrt{-4 x +1}-1}{2 \left(x^{2}+x -1\right) x}
Generating function in sympy syntax:
(-x**2*sqrt(1 - 4*x) + 2*x**2 + sqrt(1 - 4*x)/2 - 1/2)/(x*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2+x-1)^2*F(x)^2-(2*x+1)*(2*x-1)*(x^2+x-1)*F(x)+4*x^4+3*x^3-4*x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}+x -1\right)^{2} F \! \left(x \right)^{2}-\left(2 x +1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+4 x^{4}+3 x^{3}-4 x^{2}-x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(n+5) = 4*(1+2*n)/(n+6)*a(n)+6*n/(n+6)*a(1+n)-2*(13+7*n)/(n+6)*a(n+2)-(8+n)/(n+6)*a(n+3)+(24+5*n)/(n+6)*a(n+4), n >= 5
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) = 18
a \! \left(n +5\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +6}+\frac{6 n a \! \left(1+n \right)}{n +6}-\frac{2 \left(13+7 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(8+n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22046/
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[123,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[13,x,k[0]]
F[13,x,k[0]] = F[11,x]*F[14,x,k[0]]
F[14,x,k[0]] = F[119,x,k[0]]+F[15,x,k[0]]
F[15,x,k[0]] = F[16,x]+F[58,x,k[0]]
F[16,x] = F[17,x]+F[28,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[20,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[1,x]+F[11,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[11,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[11,x]*F[23,x]
F[28,x] = F[29,x]+F[35,x]
F[29,x] = F[30,x]
F[30,x] = F[11,x]*F[31,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[29,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[11,x]*F[29,x]
F[35,x] = F[36,x]+F[37,x]+F[47,x]
F[36,x] = 0
F[37,x] = F[11,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[23,x]
F[40,x] = F[41,x]
F[41,x] = F[36,x]+F[37,x]+F[42,x]
F[42,x] = F[11,x]*F[43,x]
F[43,x] = F[23,x]+F[44,x]
F[44,x] = F[36,x]+F[42,x]+F[45,x]
F[45,x] = F[11,x]*F[46,x]
F[46,x] = F[23,x]+F[44,x]
F[47,x] = F[11,x]*F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[23,x]+F[50,x]
F[50,x] = F[27,x]+F[36,x]+F[51,x]
F[51,x] = F[11,x]*F[52,x]
F[52,x] = F[11,x]+F[50,x]
F[53,x] = F[44,x]+F[54,x]
F[54,x] = 2*F[36,x]+F[55,x]+F[57,x]
F[55,x] = F[11,x]*F[56,x]
F[56,x] = F[26,x]+F[54,x]
F[57,x] = F[11,x]*F[44,x]
F[58,x,k[0]] = F[59,x,k[0]]+F[81,x,k[0]]
F[59,x,k[0]] = F[60,x,k[0]]+F[63,x,k[0]]
F[60,x,k[0]] = F[61,x,k[0]]
F[61,x,k[0]] = F[62,x,k[0]]*F[8,x,k[0]]
F[62,x,k[0]] = F[1,x]+F[60,x,k[0]]
F[63,x,k[0]] = F[36,x]+F[64,x,k[0]]+F[69,x,k[0]]
F[64,x,k[0]] = F[65,x,k[0]]*F[8,x,k[0]]
F[65,x,k[0]] = F[23,x]+F[66,x,k[0]]
F[66,x,k[0]] = F[36,x]+F[64,x,k[0]]+F[67,x,k[0]]
F[67,x,k[0]] = F[11,x]*F[68,x,k[0]]
F[68,x,k[0]] = F[60,x,k[0]]+F[66,x,k[0]]
F[69,x,k[0]] = F[11,x]*F[70,x,k[0]]
F[70,x,k[0]] = F[71,x,k[0]]+F[76,x,k[0]]
F[71,x,k[0]] = F[60,x,k[0]]+F[72,x,k[0]]
F[72,x,k[0]] = F[36,x]+F[73,x,k[0]]+F[75,x,k[0]]
F[73,x,k[0]] = F[74,x,k[0]]*F[8,x,k[0]]
F[74,x,k[0]] = F[11,x]+F[72,x,k[0]]
F[75,x,k[0]] = F[11,x]*F[60,x,k[0]]
F[76,x,k[0]] = F[66,x,k[0]]+F[77,x,k[0]]
F[77,x,k[0]] = 2*F[36,x]+F[78,x,k[0]]+F[80,x,k[0]]
F[78,x,k[0]] = F[79,x,k[0]]*F[8,x,k[0]]
F[79,x,k[0]] = F[26,x]+F[77,x,k[0]]
F[80,x,k[0]] = F[11,x]*F[66,x,k[0]]
F[81,x,k[0]] = F[82,x,k[0]]+F[92,x,k[0]]
F[82,x,k[0]] = F[36,x]+F[83,x,k[0]]+F[86,x,k[0]]
F[83,x,k[0]] = F[8,x,k[0]]*F[84,x,k[0]]
F[84,x,k[0]] = F[11,x]+F[85,x,k[0]]
F[85,x,k[0]] = F[75,x,k[0]]
F[86,x,k[0]] = F[11,x]*F[87,x,k[0]]
F[87,x,k[0]] = F[88,x,k[0]]+F[89,x,k[0]]
F[88,x,k[0]] = F[60,x,k[0]]+F[85,x,k[0]]
F[89,x,k[0]] = F[82,x,k[0]]+F[90,x,k[0]]
F[90,x,k[0]] = F[91,x,k[0]]
F[91,x,k[0]] = F[11,x]*F[82,x,k[0]]
F[92,x,k[0]] = F[107,x,k[0]]+F[36,x]+F[93,x,k[0]]+F[94,x,k[0]]
F[93,x,k[0]] = 0
F[94,x,k[0]] = F[11,x]*F[95,x,k[0]]
F[95,x,k[0]] = F[96,x,k[0]]+F[97,x,k[0]]
F[96,x,k[0]] = F[66,x,k[0]]
F[97,x,k[0]] = F[98,x,k[0]]
F[98,x,k[0]] = F[36,x]+F[93,x,k[0]]+F[94,x,k[0]]+F[99,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[104,x,k[0]]
F[101,x,k[0]] = F[102,x,k[0]]
F[102,x,k[0]] = F[103,x,k[0]]*F[11,x]
F[103,x,k[0]] = F[101,x,k[0]]+F[60,x,k[0]]
F[104,x,k[0]] = 2*F[36,x]+F[105,x,k[0]]+F[99,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[66,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[101,x,k[0]]+F[110,x,k[0]]
F[110,x,k[0]] = 2*F[36,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[72,x,k[0]]
F[113,x,k[0]] = F[101,x,k[0]]*F[11,x]
F[114,x,k[0]] = F[104,x,k[0]]+F[115,x,k[0]]
F[115,x,k[0]] = 3*F[36,x]+F[116,x,k[0]]+F[118,x,k[0]]
F[116,x,k[0]] = F[11,x]*F[117,x,k[0]]
F[117,x,k[0]] = F[115,x,k[0]]+F[77,x,k[0]]
F[118,x,k[0]] = F[104,x,k[0]]*F[11,x]
F[119,x,k[0]] = F[11,x]*F[120,x,k[0]]
F[120,x,k[0]] = F[121,x]+F[122,x,k[0]]
F[121,x] = F[1,x]+F[29,x]
F[122,x,k[0]] = F[60,x,k[0]]+F[82,x,k[0]]
F[123,x] = F[124,x]
F[124,x] = F[11,x]*F[125,x]
F[125,x] = F[126,x]+F[16,x]
F[126,x] = F[11,x]*F[121,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_{123}\! \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_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{119}\! \left(x , y\right)+F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x \right)+F_{58}\! \left(x , y\right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
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_{11}\! \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_{11}\! \left(x \right) F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right) F_{29}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{36}\! \left(x \right) = 0
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_{23}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{23}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{23}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11}\! \left(x \right) F_{44}\! \left(x \right)
F_{58}\! \left(x , y\right) = F_{59}\! \left(x , y\right)+F_{81}\! \left(x , y\right)
F_{59}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{63}\! \left(x , y\right)
F_{60}\! \left(x , y\right) = F_{61}\! \left(x , y\right)
F_{61}\! \left(x , y\right) = F_{62}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{62}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{60}\! \left(x , y\right)
F_{63}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{69}\! \left(x , y\right)
F_{64}\! \left(x , y\right) = F_{65}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{23}\! \left(x \right)+F_{66}\! \left(x , y\right)
F_{66}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{67}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{68}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{66}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{70}\! \left(x , y\right)
F_{70}\! \left(x , y\right) = F_{71}\! \left(x , y\right)+F_{76}\! \left(x , y\right)
F_{71}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{72}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{73}\! \left(x , y\right)+F_{75}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{74}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{11}\! \left(x \right)+F_{72}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{60}\! \left(x , y\right)
F_{76}\! \left(x , y\right) = F_{66}\! \left(x , y\right)+F_{77}\! \left(x , y\right)
F_{77}\! \left(x , y\right) = 2 F_{36}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{80}\! \left(x , y\right)
F_{78}\! \left(x , y\right) = F_{79}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{79}\! \left(x , y\right) = F_{26}\! \left(x \right)+F_{77}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{66}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{82}\! \left(x , y\right)+F_{92}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{83}\! \left(x , y\right)+F_{86}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{11}\! \left(x \right)+F_{85}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{75}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{87}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{88}\! \left(x , y\right)+F_{89}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{82}\! \left(x , y\right)+F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{91}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{82}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{107}\! \left(x , y\right)+F_{36}\! \left(x \right)+F_{93}\! \left(x , y\right)+F_{94}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = 0
F_{94}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{96}\! \left(x , y\right)+F_{97}\! \left(x , y\right)
F_{96}\! \left(x , y\right) = F_{66}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{98}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{93}\! \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_{11}\! \left(x \right)
F_{100}\! \left(x , y\right) = F_{101}\! \left(x , y\right)+F_{104}\! \left(x , y\right)
F_{101}\! \left(x , y\right) = F_{102}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{103}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{103}\! \left(x , y\right) = F_{101}\! \left(x , y\right)+F_{60}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = 2 F_{36}\! \left(x \right)+F_{105}\! \left(x , y\right)+F_{99}\! \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_{66}\! \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_{101}\! \left(x , y\right)+F_{110}\! \left(x , y\right)
F_{110}\! \left(x , y\right) = 2 F_{36}\! \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_{72}\! \left(x , y\right)
F_{113}\! \left(x , y\right) = F_{101}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{114}\! \left(x , y\right) = F_{104}\! \left(x , y\right)+F_{115}\! \left(x , y\right)
F_{115}\! \left(x , y\right) = 3 F_{36}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{118}\! \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_{115}\! \left(x , y\right)+F_{77}\! \left(x , y\right)
F_{118}\! \left(x , y\right) = F_{104}\! \left(x , y\right) F_{11}\! \left(x \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_{121}\! \left(x \right)+F_{122}\! \left(x , y\right)
F_{121}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{122}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{82}\! \left(x , y\right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{11}\! \left(x \right) F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{16}\! \left(x \right)
F_{126}\! \left(x \right) = F_{11}\! \left(x \right) F_{121}\! \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_123(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_13(x, k_0))
Eq(F_13(x, k_0), F_11(x)*F_14(x, k_0))
Eq(F_14(x, k_0), F_119(x, k_0) + F_15(x, k_0))
Eq(F_15(x, k_0), F_16(x) + F_58(x, k_0))
Eq(F_16(x), F_17(x) + F_28(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_1(x) + F_11(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_11(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_11(x)*F_23(x))
Eq(F_28(x), F_29(x) + F_35(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_11(x)*F_31(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_29(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_11(x)*F_29(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_47(x))
Eq(F_36(x), 0)
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_23(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_36(x) + F_37(x) + F_42(x))
Eq(F_42(x), F_11(x)*F_43(x))
Eq(F_43(x), F_23(x) + F_44(x))
Eq(F_44(x), F_36(x) + F_42(x) + F_45(x))
Eq(F_45(x), F_11(x)*F_46(x))
Eq(F_46(x), F_23(x) + F_44(x))
Eq(F_47(x), F_11(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_23(x) + F_50(x))
Eq(F_50(x), F_27(x) + F_36(x) + F_51(x))
Eq(F_51(x), F_11(x)*F_52(x))
Eq(F_52(x), F_11(x) + F_50(x))
Eq(F_53(x), F_44(x) + F_54(x))
Eq(F_54(x), 2*F_36(x) + F_55(x) + F_57(x))
Eq(F_55(x), F_11(x)*F_56(x))
Eq(F_56(x), F_26(x) + F_54(x))
Eq(F_57(x), F_11(x)*F_44(x))
Eq(F_58(x, k_0), F_59(x, k_0) + F_81(x, k_0))
Eq(F_59(x, k_0), F_60(x, k_0) + F_63(x, k_0))
Eq(F_60(x, k_0), F_61(x, k_0))
Eq(F_61(x, k_0), F_62(x, k_0)*F_8(x, k_0))
Eq(F_62(x, k_0), F_1(x) + F_60(x, k_0))
Eq(F_63(x, k_0), F_36(x) + F_64(x, k_0) + F_69(x, k_0))
Eq(F_64(x, k_0), F_65(x, k_0)*F_8(x, k_0))
Eq(F_65(x, k_0), F_23(x) + F_66(x, k_0))
Eq(F_66(x, k_0), F_36(x) + F_64(x, k_0) + F_67(x, k_0))
Eq(F_67(x, k_0), F_11(x)*F_68(x, k_0))
Eq(F_68(x, k_0), F_60(x, k_0) + F_66(x, k_0))
Eq(F_69(x, k_0), F_11(x)*F_70(x, k_0))
Eq(F_70(x, k_0), F_71(x, k_0) + F_76(x, k_0))
Eq(F_71(x, k_0), F_60(x, k_0) + F_72(x, k_0))
Eq(F_72(x, k_0), F_36(x) + F_73(x, k_0) + F_75(x, k_0))
Eq(F_73(x, k_0), F_74(x, k_0)*F_8(x, k_0))
Eq(F_74(x, k_0), F_11(x) + F_72(x, k_0))
Eq(F_75(x, k_0), F_11(x)*F_60(x, k_0))
Eq(F_76(x, k_0), F_66(x, k_0) + F_77(x, k_0))
Eq(F_77(x, k_0), 2*F_36(x) + F_78(x, k_0) + F_80(x, k_0))
Eq(F_78(x, k_0), F_79(x, k_0)*F_8(x, k_0))
Eq(F_79(x, k_0), F_26(x) + F_77(x, k_0))
Eq(F_80(x, k_0), F_11(x)*F_66(x, k_0))
Eq(F_81(x, k_0), F_82(x, k_0) + F_92(x, k_0))
Eq(F_82(x, k_0), F_36(x) + F_83(x, k_0) + F_86(x, k_0))
Eq(F_83(x, k_0), F_8(x, k_0)*F_84(x, k_0))
Eq(F_84(x, k_0), F_11(x) + F_85(x, k_0))
Eq(F_85(x, k_0), F_75(x, k_0))
Eq(F_86(x, k_0), F_11(x)*F_87(x, k_0))
Eq(F_87(x, k_0), F_88(x, k_0) + F_89(x, k_0))
Eq(F_88(x, k_0), F_60(x, k_0) + F_85(x, k_0))
Eq(F_89(x, k_0), F_82(x, k_0) + F_90(x, k_0))
Eq(F_90(x, k_0), F_91(x, k_0))
Eq(F_91(x, k_0), F_11(x)*F_82(x, k_0))
Eq(F_92(x, k_0), F_107(x, k_0) + F_36(x) + F_93(x, k_0) + F_94(x, k_0))
Eq(F_93(x, k_0), 0)
Eq(F_94(x, k_0), F_11(x)*F_95(x, k_0))
Eq(F_95(x, k_0), F_96(x, k_0) + F_97(x, k_0))
Eq(F_96(x, k_0), F_66(x, k_0))
Eq(F_97(x, k_0), F_98(x, k_0))
Eq(F_98(x, k_0), F_36(x) + F_93(x, k_0) + F_94(x, k_0) + F_99(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_104(x, k_0))
Eq(F_101(x, k_0), F_102(x, k_0))
Eq(F_102(x, k_0), F_103(x, k_0)*F_11(x))
Eq(F_103(x, k_0), F_101(x, k_0) + F_60(x, k_0))
Eq(F_104(x, k_0), F_105(x, k_0) + 2*F_36(x) + F_99(x, k_0))
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_66(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_101(x, k_0) + F_110(x, k_0))
Eq(F_110(x, k_0), F_111(x, k_0) + F_113(x, k_0) + 2*F_36(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_72(x, k_0))
Eq(F_113(x, k_0), F_101(x, k_0)*F_11(x))
Eq(F_114(x, k_0), F_104(x, k_0) + F_115(x, k_0))
Eq(F_115(x, k_0), F_116(x, k_0) + F_118(x, k_0) + 3*F_36(x))
Eq(F_116(x, k_0), F_11(x)*F_117(x, k_0))
Eq(F_117(x, k_0), F_115(x, k_0) + F_77(x, k_0))
Eq(F_118(x, k_0), F_104(x, k_0)*F_11(x))
Eq(F_119(x, k_0), F_11(x)*F_120(x, k_0))
Eq(F_120(x, k_0), F_121(x) + F_122(x, k_0))
Eq(F_121(x), F_1(x) + F_29(x))
Eq(F_122(x, k_0), F_60(x, k_0) + F_82(x, k_0))
Eq(F_123(x), F_124(x))
Eq(F_124(x), F_11(x)*F_125(x))
Eq(F_125(x), F_126(x) + F_16(x))
Eq(F_126(x), F_11(x)*F_121(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, 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": [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, 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