0132_0213_0321_1320_3021

Counting sequence:
1, 1, 2, 6, 19, 61, 198, 650, 2159, 7257, 24682, 84911, 295297, 1037416, 3678715, 13156054, 47411462, 172039802, 628127614, 2305979710, 8507419766, 31524711728, 117278718838, 437855901556, 1639979419484, 6160435543676, 23202649373988, 87602897080740, 331488047014884, 1256926707196208, 4775047830212660, 18172377749694505, 69272215182422934, 264467475657126631, 1011133992456054789, 3871060130993632926, 14838876385962855223, 56949549183749917155, 218810817750149277534, 841608296084897096365, 3240332723165120122011, 12487778081678836678008, 48169929743415123686833, 185970037801214350433511, 718568725461777453638120, 2778660592292546707791849, 10752988439903285226198667, 41642328058163331833602122, 161376230128249281873403769, 625792460294075649180101037, 2428261869455625636082831882, 9428098335677869959537104541, 36627366387319982344225999697, 142373942785775266243729120206, 553718899522157236983194167729, 2154633961217219769856612422441, 8388315498203825900054080954838, 32672723839852127083886835713745, 127320371595840050667468399189109, 496369910921427802357157290609782, 1935982260960077049965842559201957, 7554041927010452841428419783829961, 29487199316554581433943178354990022, 115148449269458102072176286283132966, 449829242177411698549211485461362075, 1757912041903790643586791706096938009, 6872291088393147279545627741374094902, 26875478637853069791171876803022592047, 105137256927377420900067816900922230089, 411432472701527346213975550591813297320, 1610563610064759533013820913026973549371, 6306532961898441951890168437823601682743, 24702060108242214048308746741093734464708, 96783605408069589564295194101033078951481, 379308947111256541984346411320276111814875, 1486975234152923640487312362576028522971444, 5830832479041888497594767301816405793805397, 22870229447553472317180713640792064930203267, 89726489155447941010660214054734211578438924, 352109873813077643940436041547134439737999015, 1382102953928391481298809553212158920988666571, 5426310466468938265532902976640454671253598078, 21309262467024911246508581756115796092004625603, 83700743120512300642601955367640880205556091471, 328840264566800078255745872567800780789171779510, 1292210282649536805427521034581069515909019222459, 5078924450250303800679688011794222982038669549507, 19966345087941774862082673793138716115734532687082, 78507587042454704233886927627821263314601195216559, 308751422255465244846179927332253948813306999996075, 1214475326783819251403185057541249869153501849164666, 4778031305886934442040937948002902869506941701722523, 18801308914217568989453193533028837170784081523668703, 73995322971798529427231932486104926858321301753262386, 291270091558217556021027343733830406198579326757705255, 1146730466196732162484633766461555017339744412104944629, 4515429808541802703776758438912215961031029987207569982, 17783106298295244644876119513704264842012241924751245617, 70046345496368471473116408253030829933306575953860901169, 275950545488494596963763299510262575435312883565902182430, 1087285461815761482684260340380876679593386011727481635421

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x^2-3*x+1)^2*F(x)^2+(x^2-3*x+1)*(2*x^3-2*x^2+3*x-1)*F(x)+x^5+2*x^4-9*x^3+12*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 61
a(n+5) = 4*(1+2*n)/(n+6)*a(n)-2*(23+19*n)/(n+6)*a(1+n)+(136+57*n)/(n+6)*a(n+2)-2*(65+18*n)/(n+6)*a(n+3)+2*(24+5*n)/(n+6)*a(n+4), n >= 6

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) = 19
a \! \left(5\right) = 61
a \! \left(n +5\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{2 \left(23+19 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(136+57 n \right) a \! \left(n +2\right)}{n +6}-\frac{2 \left(65+18 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21606/
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[9,x]
F[4,x] = F[10,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[8,x] = F[5,x]^2*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = F[13,x]+F[23,x]
F[13,x] = F[14,x]*F[5,x]
F[14,x] = F[15,x]+F[5,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[20,x]*F[9,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[20,x]*F[9,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[7,x]^2*F[5,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_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \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_{8}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{23}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{5}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{9}\! \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_{20}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{7} \left(x \right)^{2} F_{5}\! \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_9(x))
Eq(F_4(x), F_10(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))
Eq(F_8(x), F_5(x)**2*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), F_13(x) + F_23(x))
Eq(F_13(x), F_14(x)*F_5(x))
Eq(F_14(x), F_15(x) + F_5(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_20(x)*F_9(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_20(x)*F_9(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_5(x)*F_7(x)**2)
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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "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, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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"}}, {"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, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [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, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [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}}, {"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, 2, 1], "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, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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