0213_0231_0312_1032_1302_3021_3120
Counting sequence:
1, 1, 2, 6, 17, 43, 103, 239, 543, 1215, 2687, 5887, 12799, 27647, 59391, 126975, 270335, 573439, 1212415, 2555903, 5373951, 11272191, 23592959, 49283071, 102760447, 213909503, 444596223, 922746879, 1912602623, 3959422975, 8187281407, 16911433727, 34896609279, 71940702207, 148176371711, 304942678015, 627065225215, 1288490188799, 2645699854335, 5428838662143, 11132555231231, 22814866276351, 46729244180479, 95657511616511, 195713069744127, 400222232510463, 818036651065343, 1671257674219519, 3412884092616703, 6966505673588735, 14214486323888127, 28991922601197567, 59109745109237759, 120471290032160767, 245446179691692031, 499899558638125055, 1017813515785732095, 2071655828590428159, 4215369251218784255, 8574853690513424383, 17437937757178560511, 35452336266660544511, 72057594037927935999, 146421031085069565951, 297453748188566519807, 604130868413987815423, 1226708480901685182463, 2490310449950789468159, 5054407876196417142783, 10256389704982510698495, 20807927315144374222847, 42206150440647454097407, 85592892502012319498239, 173546968245459461603327, 351816302973788568420351, 713077338913316427268095, 1445044143758111435390975, 2927867219379180032491519, 5931292302484274388402175, 12013700332420377423642623, 24329632119744412140961791, 49263727149296138869276671, 99736380118206906913259519, 201890611875643072175931391, 408616927029744661050687487, 826905260616406355499024383, 1673153334346646777793347583, 3384992294920961689177292799, 6847355842297259645535780863, 13849454189505191825433952255, 28008393388831728719592685567, 56635756797306147576634933247, 114509453633897675428168990719, 231494787346366111406136229887, 467941334849873743911868956671, 945786190014030530022930907135, 1911379420656627144444247801855, 3862372922570386457685267578879, 7803974007655037252964079108095, 15766404340338603181115246116863, 31849721330734263712604668035071
Generating function in Maple syntax:
(2*x^5+x^4-5*x^2+4*x-1)/(x-1)/(2*x-1)^2
Generating function in latex syntax:
\frac{2 x^{5}+x^{4}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**5 + x**4 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^2*F(x)+2*x^5+x^4-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+2 x^{5}+x^{4}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-1+1/8*(2*n+1)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -1+\frac{\left(2 n +1\right) 2^{n}}{8} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 43
a(n+2) = -4*a(n)+4*a(n+1)-1, 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) = 17
a \! \left(5\right) = 43
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23082/
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[35,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[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[29,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[8,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[20,x]+F[21,x]+F[27,x]
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[15,x]+F[26,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[8,x]
F[34,x] = F[15,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[38,x]+F[48,x]
F[38,x] = F[39,x]+F[47,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[17,x]*F[41,x]*F[5,x]
F[41,x] = F[1,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[25,x]*F[41,x]*F[8,x]
F[44,x] = F[45,x]
F[45,x] = F[25,x]*F[41,x]*F[46,x]*F[8,x]
F[46,x] = F[21,x]
F[47,x] = F[41,x]*F[46,x]
F[48,x] = F[25,x]*F[49,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[51,x]*F[8,x]
F[51,x] = F[1,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[17,x]*F[51,x]*F[8,x]
F[54,x] = F[55,x]
F[55,x] = F[15,x]*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_{35}\! \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_{18}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right) F_{41}\! \left(x \right) F_{5}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right) F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{25}\! \left(x \right) F_{41}\! \left(x \right) F_{46}\! \left(x \right) F_{8}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right) F_{46}\! \left(x \right)
F_{48}\! \left(x \right) = F_{25}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{1}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right) F_{51}\! \left(x \right) F_{8}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \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_35(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_18(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_29(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_8(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_20(x) + F_21(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_15(x) + F_26(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_8(x))
Eq(F_34(x), F_15(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_38(x) + F_48(x))
Eq(F_38(x), F_39(x) + F_47(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_17(x)*F_41(x)*F_5(x))
Eq(F_41(x), F_1(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_25(x)*F_41(x)*F_8(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_25(x)*F_41(x)*F_46(x)*F_8(x))
Eq(F_46(x), F_21(x))
Eq(F_47(x), F_41(x)*F_46(x))
Eq(F_48(x), F_25(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_51(x)*F_8(x))
Eq(F_51(x), F_1(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_17(x)*F_51(x)*F_8(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_15(x)*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.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, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, 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