0123_0132_1203_1230_2013_3012
Counting sequence:
1, 1, 2, 6, 18, 53, 158, 486, 1550, 5109, 17298, 59799, 210048, 746983, 2682618, 9711214, 35390422, 129710309, 477769754, 1767525315, 6564644688, 24467315575, 91484660770, 343063807931, 1289912535908, 4861963178643, 18367386626558, 69533618024841, 263748085968060, 1002242485086795, 3814987038963186, 14544637112968702, 55534067024531814, 212336134707210373, 812944050739665322, 3116285512087170411, 11959798420220191824, 45950804393341219063, 176733862924445654834, 680425372004853707295, 2622127042826247922668, 10113918592737409761755, 39044429914103467214750, 150853479209483398171761, 583300119601792786110204, 2257117854094840259298091, 8740328711568357762135106, 33868773757261415630607107, 131327898242310102966347180, 509552245179898613031319179, 1978261657756723603577195718, 7684785670515442285137658729, 29869166945774877749956102708, 116157871455786937850181216323, 451959718027962478646864250362, 1759414616608836885390989357901, 6852456927844909526346677428224, 26700952856774923961839258840543, 104088460289122448148686394667894, 405944995127577273961019595078797, 1583850964596120619147525082462324, 6182127958584856803408585454063251, 24139737743045629131554467759967202, 94295850558771984399621403373767966, 368479169875816668702381079568322694, 1440418573150919687319233967953416901, 5632681584560312771887403853268248266, 22033725021956517537145528908895156347, 86218923998960285873759593253377521360, 337485502510215975851931698634411450487, 1321422108420282271080237987548935196242, 5175569924646105560599531814712477019703, 20276890389709399865290181809689463632476, 79463489365077377845930604115218914098123, 311496878311103321146981024484548424454558, 1221395654430378811847650187939440711645729, 4790408930363303911366165140257821622734012, 18793142726809884575286919142813459516669099, 73745243611532458459841267582099157886173794, 289450081175264899454586077484394424558068695, 1136359577947336271932237339914474771254966948, 4462290049988320482464450223325747812674360835, 17526585015616776834737558369554884894745646550, 68854441132780194707892887737947105659684030721, 270557451039395118028652134695725483454068929764, 1063353702922273835973056000856590292789898703251, 4180080073556524734514734513797135126562341911866, 16435314834665426797069222332015341479634771590117, 64633260585762914370496792228651092135215897651440, 254224158304000796523953750263851468431616096807567, 1000134600800354781929399869506561507052598539512822, 3935312233584004685417854810703388795154306579147933, 15487357822491889407128329439658421802774480926083956, 60960876535340415751462568532589806033490828503934803, 239993345518077005168915786526997037289323026681223298, 944973797977428207852605890261980225403315156620891539, 3721443204405954385563870580993327916841675494150669372, 14657929356129575437016877925885195275977292544443705339, 57743358069601357782187700766499181362549406811844511574, 227508830794229349661819540712601504013099032975716848585, 896519947090131496687170070707925932534952222287097511908
Generating function in Maple syntax:
1/4*((-2*x^3+5*x^2-4*x+1)*(1-4*x)^(1/2)-2*x^4+2*x^3-5*x^2+4*x-1)/(x-1/2)/x/(x-1)^2
Generating function in latex syntax:
\frac{\left(-2 x^{3}+5 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{4}+2 x^{3}-5 x^{2}+4 x -1}{4 \left(x -\frac{1}{2}\right) x \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**4/2 + x**3/2 - 5*x**2/4 + x + sqrt(1 - 4*x)*(-2*x**3 + 5*x**2 - 4*x + 1)/4 - 1/4)/(x*(x - 1)**2*(x - 1/2))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^4*F(x)^2+(2*x-1)*(2*x^4-2*x^3+5*x^2-4*x+1)*(x-1)^2*F(x)+x^7+2*x^6-15*x^5+37*x^4-43*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(2 x^{4}-2 x^{3}+5 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+2 x^{6}-15 x^{5}+37 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 53
a(6) = 158
a(n+3) = 8*(2*n+1)/(4+n)*a(n)-4*(5*n+8)/(4+n)*a(1+n)+2*(4*n+11)/(4+n)*a(n+2)+n*(3*n-13)/(4+n), n >= 7
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(5\right) = 53
a \! \left(6\right) = 158
a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{4+n}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{4+n}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{4+n}+\frac{n \left(3 n -13\right)}{4+n}, \quad n \geq 7
Specification 1
Strategy pack name: requirement_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22095/
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[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[37,x]+F[6,x]
F[6,x] = F[20,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[4,x]
F[10,x] = F[11,x]+F[12,x]
F[11,x] = F[1,x]+F[4,x]
F[12,x] = F[13,x]+F[8,x]
F[13,x] = F[14,x]+F[15,x]+F[19,x]
F[14,x] = 0
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[13,x]
F[19,x] = F[4,x]*F[8,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[14,x]+F[25,x]+F[27,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[24,x]+F[8,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[21,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]+F[27,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[21,x]+F[34,x]
F[34,x] = F[14,x]+F[32,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[21,x]+F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x,1]
F[39,x,k[0]] = F[40,x,k[0]]
F[40,x,k[0]] = F[41,x,k[0]]
F[41,x,k[0]] = F[4,x]*F[42,x,k[0]]*F[46,x,k[0]]
F[42,x,k[0]] = F[1,x]+F[43,x,k[0]]
F[43,x,k[0]] = F[44,x,k[0]]
F[44,x,k[0]] = F[42,x,k[0]]*F[45,x,k[0]]
F[45,x,k[0]] = k[0]*x
F[46,x,k[0]] = F[47,x,k[0]]+F[56,x,k[0]]
F[47,x,k[0]] = F[48,x,k[0]]+F[51,x,k[0]]
F[48,x,k[0]] = F[21,x]+F[49,x,k[0]]
F[49,x,k[0]] = F[50,x,k[0]]
F[50,x,k[0]] = F[45,x,k[0]]*F[48,x,k[0]]
F[51,x,k[0]] = F[34,x]+F[52,x,k[0]]
F[52,x,k[0]] = 2*F[14,x]+F[53,x,k[0]]+F[55,x,k[0]]
F[53,x,k[0]] = F[4,x]*F[54,x,k[0]]
F[54,x,k[0]] = F[49,x,k[0]]+F[52,x,k[0]]
F[55,x,k[0]] = F[45,x,k[0]]*F[51,x,k[0]]
F[56,x,k[0]] = 1/(-1+k[0])*(F[40,x,k[0]]*k[0]-F[40,x,1])
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_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{12}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{14}\! \left(x \right) = 0
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right)+F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x , 1\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{4}\! \left(x \right) F_{42}\! \left(x , y\right) F_{46}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{43}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{44}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{42}\! \left(x , y\right) F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = y x
F_{46}\! \left(x , y\right) = F_{47}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{48}\! \left(x , y\right)+F_{51}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{50}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{45}\! \left(x , y\right) F_{48}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{34}\! \left(x \right)+F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = 2 F_{14}\! \left(x \right)+F_{53}\! \left(x , y\right)+F_{55}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{4}\! \left(x \right) F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{49}\! \left(x , y\right)+F_{52}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{45}\! \left(x , y\right) F_{51}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = \frac{y F_{40}\! \left(x , y\right)-F_{40}\! \left(x , 1\right)}{-1+y}
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))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_37(x) + F_6(x))
Eq(F_6(x), F_20(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_4(x))
Eq(F_10(x), F_11(x) + F_12(x))
Eq(F_11(x), F_1(x) + F_4(x))
Eq(F_12(x), F_13(x) + F_8(x))
Eq(F_13(x), F_14(x) + F_15(x) + F_19(x))
Eq(F_14(x), 0)
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_13(x))
Eq(F_19(x), F_4(x)*F_8(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_14(x) + F_25(x) + F_27(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_24(x) + F_8(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_21(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x) + F_27(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_21(x) + F_34(x))
Eq(F_34(x), F_14(x) + F_32(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_21(x) + F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x, 1))
Eq(F_39(x, k_0), F_40(x, k_0))
Eq(F_40(x, k_0), F_41(x, k_0))
Eq(F_41(x, k_0), F_4(x)*F_42(x, k_0)*F_46(x, k_0))
Eq(F_42(x, k_0), F_1(x) + F_43(x, k_0))
Eq(F_43(x, k_0), F_44(x, k_0))
Eq(F_44(x, k_0), F_42(x, k_0)*F_45(x, k_0))
Eq(F_45(x, k_0), k_0*x)
Eq(F_46(x, k_0), F_47(x, k_0) + F_56(x, k_0))
Eq(F_47(x, k_0), F_48(x, k_0) + F_51(x, k_0))
Eq(F_48(x, k_0), F_21(x) + F_49(x, k_0))
Eq(F_49(x, k_0), F_50(x, k_0))
Eq(F_50(x, k_0), F_45(x, k_0)*F_48(x, k_0))
Eq(F_51(x, k_0), F_34(x) + F_52(x, k_0))
Eq(F_52(x, k_0), 2*F_14(x) + F_53(x, k_0) + F_55(x, k_0))
Eq(F_53(x, k_0), F_4(x)*F_54(x, k_0))
Eq(F_54(x, k_0), F_49(x, k_0) + F_52(x, k_0))
Eq(F_55(x, k_0), F_45(x, k_0)*F_51(x, k_0))
Eq(F_56(x, k_0), (k_0*F_40(x, k_0) - F_40(x, 1))/(k_0 - 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "limited_insertion": true, "maxreqlen": 2, "strategy_class": "RequirementInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "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.requirement_insertion", "ignore_parent": true, "strategy_class": "RequirementCorroborationFactory"}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "requirement_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": [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]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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