01342_01423_01432_10342_10423_10432_13042_13402_13420_14023_14032_14203_14230_14302_14320
Counting sequence:
1, 1, 2, 6, 24, 105, 480, 2254, 10776, 52182, 255120, 1256596, 6226176, 30998994, 154959938, 777256770, 3909932896, 19718022645, 99657159552, 504651052072, 2559864422800, 13004884066440, 66159530052000, 336990093649496, 1718426587293696, 8771842761180250, 44818676318109798, 229193577222650154, 1172983347749246712, 6007603716643871923, 30789850908737969760, 157902877149598708590, 810270475506497271648, 4160166245042665769142, 21370605947083391584064, 109833454250805158921040, 564743851788376463127552, 2905065587559026564798712, 14949869398952936391779672, 76963748136041898461910840, 396363334374486971897413440, 2041977623696161943770665276, 10523280262963291142313937152, 54248398181588297190907428592, 279738217143748425815308179360, 1442911598218745396297937814320, 7444662904955296926673260551488, 38420452532074668865166006031088, 198329409379882632831100764635136, 1024033966256863736603581800303978, 5288589916122654035241682381587950, 27318676969216976156124372199590330, 141146420927684489513408450006686216, 729401979044101825246851730108706463, 3770053343209480895768164758447683232, 19489844164751293269506335898210700490, 100773548749050183277500600923089544440, 521145771753032223705547432786105907682, 2695526980306468753751223484713852644688, 13944322139388225158890217173448537822436, 72146951267220433738518066037639775765760, 373338801518354510589492233855705186244578, 1932193755858826492349857703976485411945538, 10001348446887375934039176538543151171793618, 51775561836755708824175291984422326954249600, 268069616616721895600397007206443568426485285, 1388113885076758859606989376107033620363308544, 7188787394711895836700546573808391492376405408, 37233820321254206355045858615996907083677260608, 192872109096498077713888958837157002963541576480, 999193627631249103502085360675100700378478922880, 5176984497390582869657027623124156572463902340832, 26825616070990774155815087280695897829048244641792, 139016670329079840274756603249409070705972786593944, 720488673015960889722014884416872426535938847733672, 3734473156876173827448768996502993269403654739194200, 19358526438696300695219434908264665120544114136823712, 100358676418282978511918493022526597111373322490275220, 520326779937849377722406058952491008885969599571080320, 2697957146168839363532463962853089525815179689243213800, 13990413812144237857413908965719853035539155550772436800, 72554070859245088712336205346544589354423902169706603080, 376294458192473289093288387390486237959065726561095838080, 1951766269766116527172341054652572940888047531144861712480, 10124204517422116071533784264112133063592340939272283084800, 52520194277865530480888841894502194244096386156452160728400, 272472876859810650965410585760241882582095787356383365089040, 1413679704712466101662678577497995024147322442637964191401680, 7335147336034762443038146990304180029602046172470525811303552, 38062388472026152908428216224662486853463396661841169209650792, 197520392155014751919042096503224619684979156583505098397248000, 1025075605022951599047322535521694520425884581386736947783106912, 5320191777661419627433032310913891523899297215062146530607671616, 27613757166471227471252804694117606110230286972801667463210628064, 143334241686705976332526782778675725206511659390310238905489072768, 744046495802850725483035053822974751092774394648462978709427995360, 3862560722128054530570574956555402685196905904783698840207317663744, 20052806090269766199968142406284686262468220330372285063015975882634, 104111586307686779645879598397833228087807212953780512163926834005502, 540563290311997908371456075428471404487770975895483466731534406468058, 2806836579361414959041829577263982490567579457933434634343619236620200
Implicit equation for the generating function in Maple syntax:
(3*x^3+4*x^2+20*x-4)*F(x)^3-(x+3)*(3*x^3+4*x^2+20*x-4)*F(x)^2+(3*x^5+10*x^4+35*x^3+44*x^2+53*x-12)*F(x)-3*x^5-7*x^4-19*x^3-17*x^2-17*x+4 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{3}+4 x^{2}+20 x -4\right) F \! \left(x \right)^{3}-\left(x +3\right) \left(3 x^{3}+4 x^{2}+20 x -4\right) F \! \left(x \right)^{2}+\left(3 x^{5}+10 x^{4}+35 x^{3}+44 x^{2}+53 x -12\right) F \! \left(x \right)-3 x^{5}-7 x^{4}-19 x^{3}-17 x^{2}-17 x +4 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 105
a(6) = 480
a(n+6) = -3/2*(n-1)*(n+1)/(2*n+11)/(n+5)*a(n)-1/2*n*(67*n+191)/(2*n+11)/(n+5)*a(n+1)-5/2*(19*n+33)*(n+1)/(2*n+11)/(n+5)*a(n+2)-1/2*(401*n^2+1569*n+1480)/(2*n+11)/(n+5)*a(n+3)+2*(37*n^2+209*n+285)/(2*n+11)/(n+5)*a(n+4)+(4*n^2+55*n+165)/(2*n+11)/(n+5)*a(n+5), 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) = 24
a \! \left(5\right) = 105
a \! \left(6\right) = 480
a \! \left(n +6\right) = -\frac{3 \left(n -1\right) \left(n +1\right) a \! \left(n \right)}{2 \left(2 n +11\right) \left(n +5\right)}-\frac{n \left(67 n +191\right) a \! \left(n +1\right)}{2 \left(2 n +11\right) \left(n +5\right)}-\frac{5 \left(19 n +33\right) \left(n +1\right) a \! \left(n +2\right)}{2 \left(2 n +11\right) \left(n +5\right)}-\frac{\left(401 n^{2}+1569 n +1480\right) a \! \left(n +3\right)}{2 \left(2 n +11\right) \left(n +5\right)}+\frac{2 \left(37 n^{2}+209 n +285\right) a \! \left(n +4\right)}{\left(2 n +11\right) \left(n +5\right)}+\frac{\left(4 n^{2}+55 n +165\right) a \! \left(n +5\right)}{\left(2 n +11\right) \left(n +5\right)}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24308/
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[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[12,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[6,x,k[0]]^2*F[12,x,k[0]]*F[9,x,k[0]]
F[12,x,k[0]] = k[0]*x
F[13,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_{13}\! \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_{12}\! \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_{6}\! \left(x , y\right)^{2} F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = y x
F_{13}\! \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_13(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_12(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_6(x, k_0)**2*F_9(x, k_0))
Eq(F_12(x, k_0), k_0*x)
Eq(F_13(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, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "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": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 2, 3], "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, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 2, 3], "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": [1, 4, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 2, 0], "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, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "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": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 2, 3], "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, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 2, 3], "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": [1, 4, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 2, 0], "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, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "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": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 2, 3], "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, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 2, 3], "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": [1, 4, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 2, 0], "pos": [[0, 0], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 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