0231_1230_2013
Counting sequence:
1, 1, 2, 6, 21, 75, 265, 926, 3216, 11152, 38741, 135126, 473872, 1672151, 5939232, 21234409, 76406414, 276609127, 1007127756, 3686404720, 13559319143, 50096842078, 185845357110, 691999982641, 2585422074476, 9689463961240, 36416334225815, 137220329889356, 518292858432756, 1961938109475563, 7441771490533860, 28280281613819685, 107658852651611878, 410508913038389571, 1567673732665849576, 5995245825979242000, 22958235183467458155, 88026952772231269106, 337914675656850019326, 1298625524678902928825, 4995972059983405976588, 19239380534086204717708, 74160889313180347976555, 286121825378369201088196, 1104842177916657689013912, 4269775802467943904691879, 16513878352458847961688644, 63917093443110548159964622, 247568081416193241613384461, 959552373257938630665243022, 3721574103996718393374893816, 14442984538913828765497314241, 56085236775244921736421608562, 217917049021572435179267338456, 847179052351633612698024641981, 3295273160042022765262621290490, 12824227840429623910102147451280, 49932863978940772745182204923361, 194513375599810168048002002769866, 758076290226596960966715830041852, 2955764929710069090111129698472441, 11529589523423802860210851997881126, 44992336431033375057945495926050472, 175645922800941904100625184481320126, 685972638473110663972779788884969917, 2680028076576448417520121375936741199, 10474435199390524645315529299000215029, 40952057947581937180669677617666428987, 160165894182962850431155954695435425631, 626627004135558535680569251210353205555, 2452385145622523361713112029879437594733, 9600739643047768543454606471497624830493, 37597006432358253283231767742068176034345, 147275558164331674981009877444541955821991, 577076458031294736977846679413922928715567, 2261819203102802083347426537399919961945749, 8867495651090761121872833988683586290591399, 34774388270683136846654425012956617311820467, 136405036249234642638686256555053368958730041, 535193457156030658179519248251487143543340725, 2100379994427196263382626824999741313049457213, 8244967501394470970022620616274025833033850003, 32372887003343688788425525823730392912116631535, 127137254660171556916134969151260621957008436881, 499414030766622487407580881367798400730005781627, 1962198079615959699930247458451942553257754957739, 7711110326832628792980426469361152366446474877309, 30309641291356577843023821912096062231314860845007, 119160524537225690246964082659718350961873213443151, 468564884287460887477897736834371160958856671383985, 1842853673103273075438718134308399883566744681867403, 7249263325309654256921265178559935917877034246367023, 28521804258949924513547096470159593946892686914972465, 112237622575480760906343787457955157860516924830340971, 441750013737380421278603242329301648089013388377239647, 1738960402113275116195702820318765586123162072467260819, 6846620146571619902336079795719839299321798837421658609, 26960916782896679464554279201194231350973406754174985557, 106185072763866579785044573263970892223902749845439093161, 418274345519859269425040457616978619370448017085069874365, 1647883607366985383635795489477730746730216189370190955573
Generating function in Maple syntax:
1/2*(-(x-1)^3*(-4*x+1)^(1/2)+x^3-x^2+3*x-1)/x/(x^2-3*x+1)/(x-1)
Generating function in latex syntax:
\frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}+x^{3}-x^{2}+3 x -1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)}
Generating function in sympy syntax:
(x**3/2 - x**2/2 + 3*x/2 - sqrt(1 - 4*x)*(x - 1)**3/2 - 1/2)/(x*(x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x-1)^2*(x^2-3*x+1)^2*F(x)^2-(x-1)*(x^2-3*x+1)*(x^3-x^2+3*x-1)*F(x)+x^6-6*x^5+16*x^4-22*x^3+18*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+x^{6}-6 x^{5}+16 x^{4}-22 x^{3}+18 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+4) = -2*(2*n+1)/(n+5)*a(n)+(22+17*n)/(n+5)*a(1+n)-5*(4*n+11)/(n+5)*a(n+2)+(31+8*n)/(n+5)*a(n+3)+9/(n+5), 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) = 21
a \! \left(n +4\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{\left(22+17 n \right) a \! \left(1+n \right)}{n +5}-\frac{5 \left(4 n +11\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{9}{n +5}, \quad n \geq 5
Specification 1
Strategy pack name: col_placements_tracked_fusion
Tree: http://permpal.com/tree/365/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]*F[4,x]
F[3,x] = x
F[4,x] = F[1,x]+F[108,x]+F[5,x]
F[5,x] = F[3,x]*F[6,x]
F[6,x] = F[7,x,1]
F[7,x,k[0]] = F[1,x]+F[105,x,k[0]]+F[107,x,k[0]]+F[8,x,k[0]]
F[8,x,k[0]] = F[3,x]*F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[51,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]+F[22,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[20,x,k[0]]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[3,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x,k[0]] = F[21,x,k[0]]
F[21,x,k[0]] = F[15,x,k[0]]*F[16,x,k[0]]
F[22,x,k[0]] = F[23,x,k[0]]+F[34,x,k[0]]
F[23,x,k[0]] = F[24,x]+F[32,x,k[0]]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[3,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[1,x]+F[24,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[3,x]*F[31,x]
F[31,x] = F[24,x]+F[29,x]
F[32,x,k[0]] = F[33,x,k[0]]
F[33,x,k[0]] = F[15,x,k[0]]*F[23,x,k[0]]
F[34,x,k[0]] = F[35,x]+F[49,x,k[0]]
F[35,x] = F[36,x]+F[37,x]+F[47,x]
F[36,x] = 0
F[37,x] = F[3,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[17,x]+F[35,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[3,x]*F[43,x]
F[43,x] = F[17,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = F[3,x]*F[46,x]
F[46,x] = F[35,x]+F[44,x]
F[47,x] = F[3,x]*F[48,x]
F[48,x] = F[24,x]+F[35,x]
F[49,x,k[0]] = F[50,x,k[0]]
F[50,x,k[0]] = F[15,x,k[0]]*F[34,x,k[0]]
F[51,x,k[0]] = F[52,x,k[0]]+F[68,x,k[0]]
F[52,x,k[0]] = F[53,x,k[0]]+F[58,x,k[0]]
F[53,x,k[0]] = F[17,x]+F[54,x,k[0]]
F[54,x,k[0]] = F[36,x]+F[55,x,k[0]]+F[57,x,k[0]]
F[55,x,k[0]] = F[3,x]*F[56,x,k[0]]
F[56,x,k[0]] = F[13,x,k[0]]+F[54,x,k[0]]
F[57,x,k[0]] = F[15,x,k[0]]*F[53,x,k[0]]
F[58,x,k[0]] = F[59,x]+F[64,x,k[0]]
F[59,x] = F[36,x]+F[60,x]+F[62,x]
F[60,x] = F[3,x]*F[61,x]
F[61,x] = F[17,x]+F[59,x]
F[62,x] = F[3,x]*F[63,x]
F[63,x] = F[17,x]+F[59,x]
F[64,x,k[0]] = 2*F[36,x]+F[65,x,k[0]]+F[67,x,k[0]]
F[65,x,k[0]] = F[3,x]*F[66,x,k[0]]
F[66,x,k[0]] = F[20,x,k[0]]+F[64,x,k[0]]
F[67,x,k[0]] = F[15,x,k[0]]*F[58,x,k[0]]
F[68,x,k[0]] = F[69,x,k[0]]+F[86,x,k[0]]
F[69,x,k[0]] = F[70,x]+F[82,x,k[0]]
F[70,x] = F[36,x]+F[71,x]+F[73,x]
F[71,x] = F[3,x]*F[72,x]
F[72,x] = F[24,x]+F[70,x]
F[73,x] = F[3,x]*F[74,x]
F[74,x] = F[28,x]+F[75,x]
F[75,x] = F[76,x]+F[79,x]
F[76,x] = F[77,x]
F[77,x] = F[3,x]*F[78,x]
F[78,x] = F[17,x]+F[76,x]
F[79,x] = F[80,x]
F[80,x] = F[3,x]*F[81,x]
F[81,x] = F[29,x]+F[79,x]
F[82,x,k[0]] = 2*F[36,x]+F[83,x,k[0]]+F[85,x,k[0]]
F[83,x,k[0]] = F[3,x]*F[84,x,k[0]]
F[84,x,k[0]] = F[32,x,k[0]]+F[82,x,k[0]]
F[85,x,k[0]] = F[15,x,k[0]]*F[69,x,k[0]]
F[86,x,k[0]] = F[101,x,k[0]]+F[87,x]
F[87,x] = F[36,x]+F[88,x]+F[90,x]+F[99,x]
F[88,x] = F[3,x]*F[89,x]
F[89,x] = F[35,x]+F[87,x]
F[90,x] = F[3,x]*F[91,x]
F[91,x] = F[40,x]+F[92,x]
F[92,x] = F[93,x]+F[96,x]
F[93,x] = F[94,x]
F[94,x] = F[3,x]*F[95,x]
F[95,x] = F[41,x]+F[93,x]
F[96,x] = F[97,x]
F[97,x] = F[3,x]*F[98,x]
F[98,x] = F[44,x]+F[96,x]
F[99,x] = F[100,x]*F[3,x]
F[100,x] = F[70,x]+F[87,x]
F[101,x,k[0]] = 3*F[36,x]+F[102,x,k[0]]+F[104,x,k[0]]
F[102,x,k[0]] = F[103,x,k[0]]*F[3,x]
F[103,x,k[0]] = F[101,x,k[0]]+F[49,x,k[0]]
F[104,x,k[0]] = F[15,x,k[0]]*F[86,x,k[0]]
F[105,x,k[0]] = F[106,x,k[0]]*F[3,x]
F[106,x,k[0]] = 1/(-1+k[0])*(F[7,x,k[0]]*k[0]-F[7,x,1])
F[107,x,k[0]] = F[15,x,k[0]]*F[7,x,k[0]]
F[108,x] = F[109,x]*F[3,x]
F[109,x] = F[110,x]+F[111,x]
F[110,x] = F[19,x]+F[48,x]
F[111,x] = F[100,x]+F[63,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right) F_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{108}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x , 1\right)
F_{7}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{105}\! \left(x , y\right)+F_{107}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{51}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{22}\! \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_{20}\! \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_{3}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{32}\! \left(x , y\right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{3}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{23}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{35}\! \left(x \right)+F_{49}\! \left(x , y\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_{3}\! \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_{17}\! \left(x \right)+F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{3}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{3}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{3}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{24}\! \left(x \right)+F_{35}\! \left(x \right)
F_{49}\! \left(x , y\right) = F_{50}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{34}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{52}\! \left(x , y\right)+F_{68}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)+F_{58}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{55}\! \left(x , y\right)+F_{57}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{56}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{13}\! \left(x , y\right)+F_{54}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{53}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = F_{59}\! \left(x \right)+F_{64}\! \left(x , y\right)
F_{59}\! \left(x \right) = F_{36}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{3}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{17}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{3}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{17}\! \left(x \right)+F_{59}\! \left(x \right)
F_{64}\! \left(x , y\right) = 2 F_{36}\! \left(x \right)+F_{65}\! \left(x , y\right)+F_{67}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{66}\! \left(x , y\right)
F_{66}\! \left(x , y\right) = F_{20}\! \left(x , y\right)+F_{64}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{58}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = F_{69}\! \left(x , y\right)+F_{86}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = F_{70}\! \left(x \right)+F_{82}\! \left(x , y\right)
F_{70}\! \left(x \right) = F_{36}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{3}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{24}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{3}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{28}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{3}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{17}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{3}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{29}\! \left(x \right)+F_{79}\! \left(x \right)
F_{82}\! \left(x , y\right) = 2 F_{36}\! \left(x \right)+F_{83}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{32}\! \left(x , y\right)+F_{82}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{69}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{101}\! \left(x , y\right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{36}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)+F_{99}\! \left(x \right)
F_{88}\! \left(x \right) = F_{3}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{35}\! \left(x \right)+F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{3}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{40}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{96}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{3}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{41}\! \left(x \right)+F_{93}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{3}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{44}\! \left(x \right)+F_{96}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{3}\! \left(x \right)
F_{100}\! \left(x \right) = F_{70}\! \left(x \right)+F_{87}\! \left(x \right)
F_{101}\! \left(x , y\right) = 3 F_{36}\! \left(x \right)+F_{102}\! \left(x , y\right)+F_{104}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{103}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{103}\! \left(x , y\right) = F_{101}\! \left(x , y\right)+F_{49}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{86}\! \left(x , y\right)
F_{105}\! \left(x , y\right) = F_{106}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{106}\! \left(x , y\right) = \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{-1+y}
F_{107}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{7}\! \left(x , y\right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{3}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{111}\! \left(x \right)
F_{110}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)
F_{111}\! \left(x \right) = F_{100}\! \left(x \right)+F_{63}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_1(x) + F_108(x) + F_5(x))
Eq(F_5(x), F_3(x)*F_6(x))
Eq(F_6(x), F_7(x, 1))
Eq(F_7(x, k_0), F_1(x) + F_105(x, k_0) + F_107(x, k_0) + F_8(x, k_0))
Eq(F_8(x, k_0), F_3(x)*F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_51(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0) + F_22(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_20(x, k_0))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_3(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x, k_0), F_21(x, k_0))
Eq(F_21(x, k_0), F_15(x, k_0)*F_16(x, k_0))
Eq(F_22(x, k_0), F_23(x, k_0) + F_34(x, k_0))
Eq(F_23(x, k_0), F_24(x) + F_32(x, k_0))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_3(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_1(x) + F_24(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_3(x)*F_31(x))
Eq(F_31(x), F_24(x) + F_29(x))
Eq(F_32(x, k_0), F_33(x, k_0))
Eq(F_33(x, k_0), F_15(x, k_0)*F_23(x, k_0))
Eq(F_34(x, k_0), F_35(x) + F_49(x, k_0))
Eq(F_35(x), F_36(x) + F_37(x) + F_47(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_3(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_17(x) + F_35(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_3(x)*F_43(x))
Eq(F_43(x), F_17(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_3(x)*F_46(x))
Eq(F_46(x), F_35(x) + F_44(x))
Eq(F_47(x), F_3(x)*F_48(x))
Eq(F_48(x), F_24(x) + F_35(x))
Eq(F_49(x, k_0), F_50(x, k_0))
Eq(F_50(x, k_0), F_15(x, k_0)*F_34(x, k_0))
Eq(F_51(x, k_0), F_52(x, k_0) + F_68(x, k_0))
Eq(F_52(x, k_0), F_53(x, k_0) + F_58(x, k_0))
Eq(F_53(x, k_0), F_17(x) + F_54(x, k_0))
Eq(F_54(x, k_0), F_36(x) + F_55(x, k_0) + F_57(x, k_0))
Eq(F_55(x, k_0), F_3(x)*F_56(x, k_0))
Eq(F_56(x, k_0), F_13(x, k_0) + F_54(x, k_0))
Eq(F_57(x, k_0), F_15(x, k_0)*F_53(x, k_0))
Eq(F_58(x, k_0), F_59(x) + F_64(x, k_0))
Eq(F_59(x), F_36(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_3(x)*F_61(x))
Eq(F_61(x), F_17(x) + F_59(x))
Eq(F_62(x), F_3(x)*F_63(x))
Eq(F_63(x), F_17(x) + F_59(x))
Eq(F_64(x, k_0), 2*F_36(x) + F_65(x, k_0) + F_67(x, k_0))
Eq(F_65(x, k_0), F_3(x)*F_66(x, k_0))
Eq(F_66(x, k_0), F_20(x, k_0) + F_64(x, k_0))
Eq(F_67(x, k_0), F_15(x, k_0)*F_58(x, k_0))
Eq(F_68(x, k_0), F_69(x, k_0) + F_86(x, k_0))
Eq(F_69(x, k_0), F_70(x) + F_82(x, k_0))
Eq(F_70(x), F_36(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_3(x)*F_72(x))
Eq(F_72(x), F_24(x) + F_70(x))
Eq(F_73(x), F_3(x)*F_74(x))
Eq(F_74(x), F_28(x) + F_75(x))
Eq(F_75(x), F_76(x) + F_79(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_3(x)*F_78(x))
Eq(F_78(x), F_17(x) + F_76(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_3(x)*F_81(x))
Eq(F_81(x), F_29(x) + F_79(x))
Eq(F_82(x, k_0), 2*F_36(x) + F_83(x, k_0) + F_85(x, k_0))
Eq(F_83(x, k_0), F_3(x)*F_84(x, k_0))
Eq(F_84(x, k_0), F_32(x, k_0) + F_82(x, k_0))
Eq(F_85(x, k_0), F_15(x, k_0)*F_69(x, k_0))
Eq(F_86(x, k_0), F_101(x, k_0) + F_87(x))
Eq(F_87(x), F_36(x) + F_88(x) + F_90(x) + F_99(x))
Eq(F_88(x), F_3(x)*F_89(x))
Eq(F_89(x), F_35(x) + F_87(x))
Eq(F_90(x), F_3(x)*F_91(x))
Eq(F_91(x), F_40(x) + F_92(x))
Eq(F_92(x), F_93(x) + F_96(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_3(x)*F_95(x))
Eq(F_95(x), F_41(x) + F_93(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_3(x)*F_98(x))
Eq(F_98(x), F_44(x) + F_96(x))
Eq(F_99(x), F_100(x)*F_3(x))
Eq(F_100(x), F_70(x) + F_87(x))
Eq(F_101(x, k_0), F_102(x, k_0) + F_104(x, k_0) + 3*F_36(x))
Eq(F_102(x, k_0), F_103(x, k_0)*F_3(x))
Eq(F_103(x, k_0), F_101(x, k_0) + F_49(x, k_0))
Eq(F_104(x, k_0), F_15(x, k_0)*F_86(x, k_0))
Eq(F_105(x, k_0), F_106(x, k_0)*F_3(x))
Eq(F_106(x, k_0), (k_0*F_7(x, k_0) - F_7(x, 1))/(k_0 - 1))
Eq(F_107(x, k_0), F_15(x, k_0)*F_7(x, k_0))
Eq(F_108(x), F_109(x)*F_3(x))
Eq(F_109(x), F_110(x) + F_111(x))
Eq(F_110(x), F_19(x) + F_48(x))
Eq(F_111(x), F_100(x) + F_63(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": false, "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": "col_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, 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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 2, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", 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