01324_01342_01432_03124_03142_03412_04132_04312_10324_10342_10432_14032_40132_40312_41032

Counting sequence:
1, 1, 2, 6, 24, 105, 474, 2177, 10120, 47475, 224310, 1065900, 5088816, 24389859, 117282186, 565560630, 2733921696, 13244000331, 64278928422, 312494913338, 1521476438920, 7417715164923, 36207771391638, 176934250208061, 865483086680184, 4237455779245525, 20764404202724234, 101829355282353132, 499735779412049040, 2454139570018046745, 12059497574417261970, 59294188249346966865, 291696743959196983584, 1435729770679161819411, 7070041952865247073574, 34831094731760316692010, 171670967658559470363624, 846449151305604321072027, 4175127201790805865425550, 20601301113484046778936555, 101687517465999158466406200, 502089637499286893924337705, 2479868192780619580363003650, 12251895055311734469174780540, 60547932613801268377745147760, 299303095857143914882052150625, 1479904762851015680900713295910, 7319175546210046002957403416540, 36206944288344731226552826487040, 179150762050820362896477101238885, 886622082574198551682691339389050, 4388820179921509609169101331168670, 21729156715689669361351064901437976, 107602034964858970922243877140423109, 532938995703173346761909493044030826, 2640042777249884777398645856887481675, 13080311571804656967009562251510258856, 64818090729293617992825267933874726299, 321249824866136675712765207247450748694, 1592412997717979744014875188019327738876, 7894641751332706859845516590799470935280, 39144513735609296485598291979115689447451, 194119475184101409736497273871639050657962, 962775681686957318031847941810267220909686, 4775700387107577893701574275413227628161664, 23692082492163959156453521803127958417132595, 117549809479336144207074663683848700762060454, 583299435866491334966487795648595221143734674, 2894746861222381462572131979586875778452533448, 14367379983807911145964078506547045951331817867, 71316680672587844658580370561980838200449567630, 354038007488784572994574186183454330525251722507, 1757731760265293159371584299005740200636734927512, 8727664308542563386884923862838617723685019989529, 43339611622721811085618154593790102224068261510626, 215234764070308054578504724291095739650329157756900, 1069003588606576614238575058562324554964257699427856, 5309873939696556298960982215474151509255245637532361, 26377070733237841979990250349328977630849800317992062, 131040416192582184341595494398142498230923366146633258, 651057562128694215893326041829213109301002726425005280, 3234953540371592872760652911750569300081462651620543153, 16074978079137188467709282869485485781030667565212630098, 79885064244747547843131989270505015581709760796109974286, 397020427287946501618356014961312606021117442734385415064, 1973292271753469364144344920341546910007825873155676393505, 9808453518516763062221233426907394427851558638687422573314, 48757284512993339662276539976628475392768669261034008595999, 242386061719501356418679190310199205726886890800097234665448, 1205047791158290318938070867857830875843907714551212437508447, 5991406057727831583462201994691764887835355594460704784746910, 29790683774092352227320495481499996815742641320317474887601892, 148135389293731425502003138356142711849966360079760167939815152, 736653452703278151379400937770371413981305066164580143422012579, 3663474110949506021082154995693922795211547629472561408444423966, 18219983267717391730419813520614069308728368701062939237213773165, 90620664691377502941883002150741016057582811550711378348741632736, 450744489028025997554472779744064880573856073288251760647444918301, 2242110571714771952656623856849082140922432517719344828483358782842, 11153381743800451314615336415016827524620994576060300110761000282998, 55485382306979689739887543092601444303437911452138483156853576664600

Generating function in Maple syntax:
((-7*x^2+5*x)*(5*x^2-6*x+1)^(1/2)+10*x^4-37*x^3+72*x^2-53*x+8)/(10*x^4-52*x^3+90*x^2-56*x+8)

Generating function in latex syntax:
\frac{\left(-7 x^{2}+5 x \right) \sqrt{5 x^{2}-6 x +1}+10 x^{4}-37 x^{3}+72 x^{2}-53 x +8}{10 x^{4}-52 x^{3}+90 x^{2}-56 x +8}

Generating function in sympy syntax:
(10*x**4 - 37*x**3 + 72*x**2 - 53*x + (-7*x**2 + 5*x)*sqrt(5*x**2 - 6*x + 1) + 8)/(10*x**4 - 52*x**3 + 90*x**2 - 56*x + 8)

Implicit equation for the generating function in Maple syntax:
(5*x-1)*(x-1)*(x-2)^2*F(x)^2-(5*x-1)*(x-1)*(2*x^2-5*x+8)*F(x)+5*x^4-11*x^3+26*x^2-25*x+4 = 0

Implicit equation for the generating function in latex syntax:
\left(5 x -1\right) \left(x -1\right) \left(x -2\right)^{2} F \! \left(x \right)^{2}-\left(5 x -1\right) \left(x -1\right) \left(2 x^{2}-5 x +8\right) F \! \left(x \right)+5 x^{4}-11 x^{3}+26 x^{2}-25 x +4 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(n+4) = -7/2*(n+1)/(3+n)*a(n)+1/10*(184+79*n)/(3+n)*a(3+n)+1/10*(138+137*n)/(3+n)*a(n+1)-9/10*(29+19*n)/(3+n)*a(n+2), 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) = 24
a \! \left(n +4\right) = -\frac{7 \left(n +1\right) a \! \left(n \right)}{2 \left(3+n \right)}+\frac{\left(184+79 n \right) a \! \left(3+n \right)}{30+10 n}+\frac{\left(138+137 n \right) a \! \left(n +1\right)}{30+10 n}-\frac{9 \left(29+19 n \right) a \! \left(n +2\right)}{10 \left(3+n \right)}, \quad n \geq 5

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24302/
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[22,x]*F[4,x]
F[4,x] = F[5,x,1]
F[5,x,k[0]] = (F[6,x,k[0]]*k[0]-F[6,x,1])/(-1+k[0])
F[6,x,k[0]] = F[1,x]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]
F[8,x,k[0]] = F[17,x,k[0]]*F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[6,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]*F[17,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]+F[15,x,k[0]]
F[13,x,k[0]] = F[14,x,k[0]]+F[18,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]+F[6,x,k[0]]
F[15,x,k[0]] = F[16,x,k[0]]
F[16,x,k[0]] = F[12,x,k[0]]*F[17,x,k[0]]
F[17,x,k[0]] = k[0]*x
F[18,x,k[0]] = F[19,x,k[0]]
F[19,x,k[0]] = F[14,x,k[0]]*F[17,x,k[0]]*F[20,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]+F[9,x,k[0]]
F[21,x,k[0]] = F[11,x,k[0]]
F[22,x] = 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_{3}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x , 1\right)
F_{5}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{6}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = y x
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{11}\! \left(x , y\right)
F_{22}\! \left(x \right) = x
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_22(x)*F_4(x))
Eq(F_4(x), F_5(x, 1))
Eq(F_5(x, k_0), (-k_0*F_6(x, k_0) + F_6(x, 1))/(1 - k_0))
Eq(F_6(x, k_0), F_1(x) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0))
Eq(F_8(x, k_0), F_17(x, k_0)*F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_6(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0)*F_17(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0) + F_15(x, k_0))
Eq(F_13(x, k_0), F_14(x, k_0) + F_18(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0) + F_6(x, k_0))
Eq(F_15(x, k_0), F_16(x, k_0))
Eq(F_16(x, k_0), F_12(x, k_0)*F_17(x, k_0))
Eq(F_17(x, k_0), k_0*x)
Eq(F_18(x, k_0), F_19(x, k_0))
Eq(F_19(x, k_0), F_14(x, k_0)*F_17(x, k_0)*F_20(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0) + F_9(x, k_0))
Eq(F_21(x, k_0), F_11(x, k_0))
Eq(F_22(x), x)
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"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.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 1, 0, 3, 2], "pos": [[0, 0], [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, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], 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