0123_0231_2031_2301
Counting sequence:
1, 1, 2, 6, 20, 65, 199, 574, 1574, 4143, 10553, 26180, 63568, 151645, 356459, 827514, 1900682, 4325531, 9765037, 21889216, 48758996, 108003561, 238027007, 522191126, 1140850990, 2483028295, 5385486689, 11643388284, 25098715544, 53955527093, 115695682003, 247497490930, 528280977938, 1125281432115, 2392296784469, 5076651344504, 10754598109852, 22746146800321, 48034914239207, 101292508709646, 213305255789366, 448600744133471, 942281465004937, 1976921906742196, 4142959813460960, 8672947719898125, 18137543811793979, 37893568739673194, 79094468455695514, 164944336352445643, 343680946563712253, 715509390798488880, 1488439676845950308, 3093972944003532185, 6426636668257699279, 13339662096271410694, 27670116110564329022, 57357844854190638711, 118822972568543168177, 246004626045485975276, 509014844283922941736, 1052617333706051290981, 2175562879193120245667, 4494088024957489514466, 9278712269075904464930, 19147720348510514579555, 39494479061812150012069, 81423928341353960835304, 167791584094462081501484, 345618196965022159079793, 711601599387419663141303, 1464523905499948721900030, 3012869816070833646340678, 6195744825524974520371855, 12736222404299433141340889, 26171355048063573774306084, 53759420040988041112718192, 110390038903560826515360701, 226598033314017055933991947, 464983093369276746321366106, 953842471675942418843176106, 1956041976136470004674596091, 4010006943661724972500389197, 8218277721740249129652587936, 16837918815592555145307622900, 34488235781966141096017792585, 70620610678608177869635980959, 144568185212795815228063354614, 295867669389205885700890693454, 605352679210550954425671749543, 1238249524295201619967849009153, 2532206350358245352306158604380, 5177065244291460309628137509048, 10581911455811430589837713871125, 21624336606237022641937901949299, 44179604122016651250599945310674, 90240877103747080519046559437362, 184284706008178849242583228486291, 376254543780241339231740220150517, 768037807413278488631815054562136, 1567449967182205654950673513452476
Generating function in Maple syntax:
-(13*x^5-35*x^4+42*x^3-26*x^2+8*x-1)/(2*x-1)^3/(x-1)^3
Generating function in latex syntax:
-\frac{13 x^{5}-35 x^{4}+42 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right)^{3} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-13*x**5 + 35*x**4 - 42*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^3*(x-1)^3*F(x)+13*x^5-35*x^4+42*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{3} \left(x -1\right)^{3} F \! \left(x \right)+13 x^{5}-35 x^{4}+42 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
1/8*(n^2-n-8)*2^n+1/2*n^2+1/2*n+2
Explicit closed form in latex syntax:
\frac{\left(n^{2}-n -8\right) 2^{n}}{8}+\frac{n^{2}}{2}+\frac{n}{2}+2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-1/2*(-1+n)*(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) = 20
a \! \left(5\right) = 65
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{\left(-1+n \right) \left(n -4\right)}{2}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15044/
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[5,x]
F[4,x] = x
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[44,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = 2*F[21,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[25,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[40,x]
F[34,x] = F[35,x]
F[35,x] = F[21,x]+F[36,x]+F[38,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[11,x]+F[35,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[11,x]+F[35,x]
F[40,x] = F[41,x]
F[41,x] = 2*F[21,x]+F[32,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[14,x]+F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[39,x]+F[46,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[21,x]+F[44,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[47,x]+F[7,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{44}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \left(x \right)+F_{7}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_44(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), 2*F_21(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_40(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_21(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_11(x) + F_35(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_11(x) + F_35(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), 2*F_21(x) + F_32(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_14(x) + F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_39(x) + F_46(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_21(x) + F_44(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_47(x) + F_7(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.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": 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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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