0123_0231_0312_1230_3012

Counting sequence:
1, 1, 2, 6, 19, 60, 191, 619, 2048, 6909, 23704, 82489, 290500, 1033399, 3707838, 13402682, 48760351, 178405140, 656043839, 2423307028, 8987427447, 33453694466, 124936258105, 467995871754, 1757900019077, 6619846420528, 24987199492679, 94520750408682, 358268702159041, 1360510918810408, 5175497420902711, 19720133460129619, 75254198337177816, 287590328749420925, 1100534370899151688, 4216819865806452949, 16176618251666906440, 62127422576288648803, 238861285363295350202, 919286657093271150591, 3541413699369763259410, 13655332291007661393429, 52699762202912105352668, 203553241407997457013367, 786853361000994150101406, 3043971215078242223355125, 11784299926611415613401444, 45653073683802462499830933, 176980971925971827977822832, 686533217105588966032431403, 2664794874861749619656205858, 10349580545376066004887022013, 40218747491148691955029439524, 156376618946931126205583285403, 608336336974884597653192794826, 2367750953583703468645672670797, 9220207881428576966195331135108, 35921160738203428870440552047771, 140009621027325732903938870859850, 545954616154902718634582314226961, 2129805580751022761321355093265856, 8311933539335878411808435940482191, 32451671282381505237519894486755502, 126747521841153485025455279433135626, 495226691716970144504464322146682575, 1935645264867889813376954216390547924, 7568326849428202548370869922239693023, 29602051871384720011729422536296642972, 115820975870345005737915063199997751471, 453306478380560981294698856655056376170, 1774728586800843251784641033845285920769, 6950298511446948811203581227840351637118, 27227188901156348674132579796094992662817, 106690678266233726515340817428444261547316, 418187556577337047652877108947253395574555, 1639583211007715859481637830955215526365574, 6429992141371019770810024039350079987390093, 25223134868180904346021385318437625180640132, 98968378479713362805711537173084954419975731, 388418459654978262259995383202575721684367960, 1524778037602314534191628260207243178351981899, 5987068087590635016654869557713376361851636638, 23513653103207411851390010075629031998247870917, 92368094235987606559278062110297679141233077016, 362925545275382724587920525399466245561904357455, 1426279248197656560560957183442942704285007761974, 5606359321754181295075653011613850162713759076293, 22041674156419608092144797972376736306081349471232, 86674934742182522462641435458522917768762884732231, 340899093046183318986594876237364564855310256758830, 1341033693846538100915994126773906429217771346709629, 5276345927430542786333847699537255938991803026733428, 20763703749922432193462174663315599171005734154569027, 81724580285262847944924738244145248062975463062007026, 321717925803339853113840514867621971069256290550236625, 1266691723780768060966446385322561567906487048785140674, 4988134928186722446530316926701808227615993746163834973, 19646064284316297883547194773358840989328948697063590072, 77389422353917655665734895381401697323349680321820201071, 304898253148147005327554435777090551279391362923361248410, 1201418200238278502014724505851191183700228884462107157729

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

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

Generating function in sympy syntax:
(-x**2 - x/2 + sqrt(1 - 4*x)*(x - 1)/2 + 1/2)/(x*(x - 1)**2)

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+2) = -2*(2*n+3)/(3+n)*a(n)+(5*n+9)/(3+n)*a(1+n)+3*(1+n)/(3+n), n >= 4

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(n +2\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{3+n}+\frac{\left(5 n +9\right) a \! \left(1+n \right)}{3+n}+\frac{3 n +3}{3+n}, \quad n \geq 4

Specification 1
Strategy pack name: requirement_placements_tracked_fusion
Tree: http://permpal.com/tree/21532/
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[13,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[1,x]+F[11,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[8,x,k[0]]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[12,x,k[0]]*F[13,x]
F[12,x,k[0]] = (F[8,x,k[0]]*k[0]-F[8,x,1])/(-1+k[0])
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[17,x]+F[34,x]
F[17,x] = F[18,x]+F[26,x]
F[18,x] = F[19,x]*F[20,x]
F[19,x] = F[20,x]+F[23,x]+F[24,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[20,x]
F[23,x] = F[13,x]*F[19,x]
F[24,x] = F[25,x]
F[25,x] = F[13,x]*F[19,x]*F[20,x]
F[26,x] = F[20,x]*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[30,x,1]
F[30,x,k[0]] = F[31,x]+F[32,x,k[0]]+F[9,x,k[0]]
F[31,x] = 0
F[32,x,k[0]] = F[13,x]*F[33,x,k[0]]
F[33,x,k[0]] = (F[30,x,k[0]]*k[0]-F[30,x,1])/(-1+k[0])
F[34,x] = F[35,x]
F[35,x] = F[36,x,1]
F[36,x,k[0]] = F[37,x,k[0]]
F[37,x,k[0]] = F[38,x,k[0]]
F[38,x,k[0]] = F[13,x]*F[39,x,k[0]]*F[50,x,k[0]]
F[39,x,k[0]] = F[40,x,k[0]]+F[49,x,k[0]]
F[40,x,k[0]] = F[20,x]*F[41,x,k[0]]
F[41,x,k[0]] = (F[42,x,k[0]]-F[42,x,1])/(-1+k[0])
F[42,x,k[0]] = F[31,x]+F[43,x,k[0]]+F[47,x,k[0]]
F[43,x,k[0]] = F[10,x,k[0]]*F[44,x,k[0]]
F[44,x,k[0]] = F[20,x]+F[43,x,k[0]]+F[45,x,k[0]]
F[45,x,k[0]] = F[46,x,k[0]]
F[46,x,k[0]] = F[13,x]*F[20,x]*F[44,x,k[0]]
F[47,x,k[0]] = F[48,x,k[0]]
F[48,x,k[0]] = F[13,x]*F[20,x]*F[42,x,k[0]]
F[49,x,k[0]] = (F[37,x,k[0]]*k[0]-F[37,x,1])/(-1+k[0])
F[50,x,k[0]] = F[1,x]+F[51,x,k[0]]
F[51,x,k[0]] = F[52,x,k[0]]
F[52,x,k[0]] = F[10,x,k[0]]*F[50,x,k[0]]
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_{13}\! \left(x \right) F_{4}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x , 1\right)
F_{8}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{13}\! \left(x \right)
F_{12}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{34}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x , 1\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x \right)+F_{32}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x , y\right) = F_{13}\! \left(x \right) F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = \frac{y F_{30}\! \left(x , y\right)-F_{30}\! \left(x , 1\right)}{-1+y}
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x , 1\right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{38}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{13}\! \left(x \right) F_{39}\! \left(x , y\right) F_{50}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)+F_{49}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{20}\! \left(x \right) F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = \frac{F_{42}\! \left(x , y\right)-F_{42}\! \left(x , 1\right)}{-1+y}
F_{42}\! \left(x , y\right) = F_{31}\! \left(x \right)+F_{43}\! \left(x , y\right)+F_{47}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{44}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{20}\! \left(x \right)+F_{43}\! \left(x , y\right)+F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{44}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{48}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{42}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = \frac{y F_{37}\! \left(x , y\right)-F_{37}\! \left(x , 1\right)}{-1+y}
F_{50}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{51}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{50}\! \left(x , y\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_13(x)*F_4(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_13(x)*F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_1(x) + F_11(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_8(x, k_0))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_12(x, k_0)*F_13(x))
Eq(F_12(x, k_0), (-k_0*F_8(x, k_0) + F_8(x, 1))/(1 - k_0))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_34(x))
Eq(F_17(x), F_18(x) + F_26(x))
Eq(F_18(x), F_19(x)*F_20(x))
Eq(F_19(x), F_20(x) + F_23(x) + F_24(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_20(x))
Eq(F_23(x), F_13(x)*F_19(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_13(x)*F_19(x)*F_20(x))
Eq(F_26(x), F_20(x)*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_30(x, 1))
Eq(F_30(x, k_0), F_31(x) + F_32(x, k_0) + F_9(x, k_0))
Eq(F_31(x), 0)
Eq(F_32(x, k_0), F_13(x)*F_33(x, k_0))
Eq(F_33(x, k_0), (-k_0*F_30(x, k_0) + F_30(x, 1))/(1 - k_0))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x, 1))
Eq(F_36(x, k_0), F_37(x, k_0))
Eq(F_37(x, k_0), F_38(x, k_0))
Eq(F_38(x, k_0), F_13(x)*F_39(x, k_0)*F_50(x, k_0))
Eq(F_39(x, k_0), F_40(x, k_0) + F_49(x, k_0))
Eq(F_40(x, k_0), F_20(x)*F_41(x, k_0))
Eq(F_41(x, k_0), -F_42(x, k_0) + (-k_0*F_42(x, k_0) + F_42(x, 1))/(1 - k_0))
Eq(F_42(x, k_0), F_31(x) + F_43(x, k_0) + F_47(x, k_0))
Eq(F_43(x, k_0), F_10(x, k_0)*F_44(x, k_0))
Eq(F_44(x, k_0), F_20(x) + F_43(x, k_0) + F_45(x, k_0))
Eq(F_45(x, k_0), F_46(x, k_0))
Eq(F_46(x, k_0), F_13(x)*F_20(x)*F_44(x, k_0))
Eq(F_47(x, k_0), F_48(x, k_0))
Eq(F_48(x, k_0), F_13(x)*F_20(x)*F_42(x, k_0))
Eq(F_49(x, k_0), (-k_0*F_37(x, k_0) + F_37(x, 1))/(1 - k_0))
Eq(F_50(x, k_0), F_1(x) + F_51(x, k_0))
Eq(F_51(x, k_0), F_52(x, k_0))
Eq(F_52(x, k_0), F_10(x, k_0)*F_50(x, k_0))
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, 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, 2, 3, 0], "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, 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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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