0123_0132_0213_0231_0312_1023_1230

Counting sequence:
1, 1, 2, 6, 17, 51, 158, 506, 1665, 5603, 19202, 66795, 235223, 836906, 3003669, 10860980, 39527474, 144675574, 532196958, 1966497022, 7295551210, 27164194832, 101476042662, 380217287504, 1428537863840, 5380797405472, 20314741940012, 76861737189336, 291390879963704, 1106746882054048, 4210882047458688, 16047278466647341, 61247589351927318, 234096933353769571, 895952434443184543, 3433390820554524612, 13172856145950861385, 50597252905266674869, 194553167513602495290, 748841929361838269149, 2885096767421957072869, 11125730246145201367218, 42941338333833704265307, 165876117873159813333985, 641264096594904859026632, 2480957948952316052526297, 9605432903003146228345529, 37214835897613335777254996, 144279393285594800597162935, 559718576120655908951034603, 2172707381982794225117363422, 8438949970861605865675439725, 32796038628102709655704404503, 127523693108974123821791272596, 496121578309247870674551340643, 1931100169999087394041212979787, 7520264059544389891272942908770, 29299860158473455692009718336909, 114207631650365297712535759145095, 445364026647293435101690602037036, 1737477627766201789770355696853915, 6781118648402230832672354367068391, 26476206060629203797667087762979282, 103413437524757787592563368530154486, 404072962902431671081304114060300593, 1579425803702292468067348334120773479, 6175761778014117205270078259786353822, 24156231019112614401702063502114932151, 94517191774130499869799229811545444523, 369940001304498804102099446105017892874, 1448393798988695598905057059651734257297, 5672481616519017691936154522561588321921, 22222195880571007594351269956472897888568, 87081276472458003264566663543129114069289, 341336861299054954155245987386061877100597, 1338316913890788448111037081418343987933666, 5248667051122768501765908555191308195267763, 20589717761460137613759920707302332915426949, 80790485697305480000004619838053425156780272, 317085041022498058608358234720199196097263851, 1244785023491874911085711885533429310657265833, 4887800536555357180225678540755094233586570484, 19196880525675688443583289646698045997140415857, 75412537179394463496961879704206131737315958161, 296312427611685075261299580605404097012142448358, 1164521216673958256702655655063918121051890530599, 4577560014919894230505614405796753217769680888757, 17997315992702754549762719193452650023760410499976, 70772821758720136353695826124632698560844036254793, 278361035469660773056915348610780814208470283520369, 1095044299724734894816056984493113730045228037699362, 4308582967978045571361426655330702189665268911292731, 16955672856610665187887957030838809641950396396803893, 66737774537437237103583620303220817491571463087699024, 262725916376784491708924790376706701247832958065294517, 1034444143373572931496198504511390392570354351899667991, 4073639466178883600916387965812404156690463249108752758, 16044569761276077583865593775601580651264265854529481549, 63203637940926743529952804502799808380262505777495090207, 249013680426830812993450238066047386653547722608809008096, 981228434663659703548753572247258973498606067323508960283

Generating function in Maple syntax:
1/2*(2*x^3-(-4*x+1)^(1/2)*x+(-4*x+1)^(1/2)+x-1)/(x^2+x-1)/x

Generating function in latex syntax:
\frac{2 x^{3}-\sqrt{-4 x +1}\, x +\sqrt{-4 x +1}+x -1}{2 \left(x^{2}+x -1\right) x}

Generating function in sympy syntax:
(x**3 - x*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-(x^2+x-1)*(2*x^3+x-1)*F(x)+x^5+x^3-2*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(x^{2}+x -1\right) \left(2 x^{3}+x -1\right) F \! \left(x \right)+x^{5}+x^{3}-2 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(n+4) = 2*(3+2*n)/(n+5)*a(n)-(11+n)/(n+5)*a(1+n)-2*(9+4*n)/(n+5)*a(n+2)+2*(11+3*n)/(n+5)*a(n+3), 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) = 17
a \! \left(n +4\right) = \frac{2 \left(3+2 n \right) a \! \left(n \right)}{n +5}-\frac{\left(11+n \right) a \! \left(1+n \right)}{n +5}-\frac{2 \left(9+4 n \right) a \! \left(n +2\right)}{n +5}+\frac{2 \left(11+3 n \right) a \! \left(n +3\right)}{n +5}, \quad n \geq 5

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/22694/
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[8,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[8,x]
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]
F[13,x] = F[6,x]*F[8,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[5,x]*F[8,x]
F[16,x] = F[17,x,1]
F[17,x,k[0]] = (F[18,x,k[0]]*k[0]-F[18,x,1])/(-1+k[0])
F[18,x,k[0]] = F[19,x,k[0]]+F[23,x,k[0]]
F[19,x,k[0]] = F[1,x]+F[20,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]
F[21,x,k[0]] = F[19,x,k[0]]*F[22,x,k[0]]
F[22,x,k[0]] = k[0]*x
F[23,x,k[0]] = F[24,x,k[0]]
F[24,x,k[0]] = F[17,x,k[0]]*F[19,x,k[0]]*F[8,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_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x , 1\right)
F_{17}\! \left(x , y\right) = \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{23}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{19}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = y x
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{19}\! \left(x , y\right) F_{8}\! \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))
Eq(F_3(x), F_4(x)*F_8(x))
Eq(F_4(x), F_14(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_8(x))
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_6(x)*F_8(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_5(x)*F_8(x))
Eq(F_16(x), F_17(x, 1))
Eq(F_17(x, k_0), (-k_0*F_18(x, k_0) + F_18(x, 1))/(1 - k_0))
Eq(F_18(x, k_0), F_19(x, k_0) + F_23(x, k_0))
Eq(F_19(x, k_0), F_1(x) + F_20(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0))
Eq(F_21(x, k_0), F_19(x, k_0)*F_22(x, k_0))
Eq(F_22(x, k_0), k_0*x)
Eq(F_23(x, k_0), F_24(x, k_0))
Eq(F_24(x, k_0), F_17(x, k_0)*F_19(x, k_0)*F_8(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": 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": [0, 3, 1, 2], "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, 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": [0, 3, 1, 2], "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": [[{"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, 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": [0, 3, 1, 2], "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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 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