0132_0213_0231_0321_1032_3021_3120
Counting sequence:
1, 1, 2, 6, 17, 44, 108, 256, 592, 1344, 3008, 6656, 14592, 31744, 68608, 147456, 315392, 671744, 1425408, 3014656, 6356992, 13369344, 28049408, 58720256, 122683392, 255852544, 532676608, 1107296256, 2298478592, 4764729344, 9865003008, 20401094656, 42144366592, 86973087744, 179314884608, 369367187456, 760209211392, 1563368095744, 3212635537408, 6597069766656, 13537736916992, 27762668601344, 56899726737408, 116548232544256, 238594023227392, 488183162732544, 998356558020608, 2040693581152256, 4169348092526592, 8514618045497344, 17381079811883008, 35465847065542656, 72339069014638592, 147492887796383744, 300615275126980608, 612489549322387456, 1247497096781627392, 2540030189836959744, 5170132372221329408, 10520408729537478656, 21401105429264596992, 43522786798908473344, 88486725478575505408, 179855754718668128256, 365476116960370491392, 742481448966809452544, 1508021328025755844608, 3062159516235785568256, 6216552752840118894592, 12617572946417333305344, 25604080774308857643008, 51946031311566097350656, 105367802149028958830592, 213687083349851445919744, 433277124803289948356608, 878360165813754009747456, 1780332164041856245563392, 3607887992912408943263744, 7310223315482210790801408, 14809341290279207390150656, 29996471899187986397396992, 60748522435635116028985344, 123008202145788518526353408, 249038718840613609989472256, 504122066779300365852475392, 1020333391754747023452012544, 2064845299901786630398148608, 4178047632588158427784544256, 8452809330745487189545582592, 17099046792629315047044153344, 34584949847535311429994283008, 69943612219623985531800518656, 141434649488354696407224942592, 285964149074922843501697695744, 578117998346272588377891012608, 1168615397085398979504773267456, 2361989594956505564507529019392, 4773496791484426340011023007744, 9646028786111683102013975953408, 19490127978509027048011811782656, 39376396769589375783991343316992
Generating function in Maple syntax:
(x^4+2*x^3+2*x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
\frac{x^{4}+2 x^{3}+2 x^{2}-3 x +1}{\left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(x**4 + 2*x**3 + 2*x**2 - 3*x + 1)/(2*x - 1)**2
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*F(x)+x^4+2*x^3+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} F \! \left(x \right)+x^{4}+2 x^{3}+2 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/16*2^n*(5*n-3))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{2^{n} \left(5 n -3\right)}{16} & \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(n+2) = -4*a(n)+4*a(n+1), n >= 5
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(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22820/
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[33,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[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[8,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[29,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[23,x]+F[24,x]+F[27,x]
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[19,x]+F[26,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[28,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[19,x]*F[8,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[36,x]+F[45,x]
F[36,x] = F[14,x]*F[37,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]*F[42,x]
F[39,x] = F[1,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[21,x]*F[39,x]*F[8,x]
F[42,x] = F[1,x]+F[8,x]
F[43,x] = F[44,x]
F[44,x] = F[8,x]^3*F[21,x]
F[45,x] = F[46,x]
F[46,x] = F[14,x]*F[47,x]*F[50,x]*F[8,x]
F[47,x] = F[37,x]+F[48,x]
F[48,x] = F[40,x]+F[49,x]
F[49,x] = x^2
F[50,x] = F[14,x]+F[51,x]
F[51,x] = F[19,x]+F[52,x]
F[52,x] = F[23,x]+F[53,x]+F[55,x]
F[53,x] = F[54,x]*F[8,x]
F[54,x] = F[15,x]+F[52,x]
F[55,x] = F[51,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_{33}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \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_{19}\! \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_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{45}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{21}\! \left(x \right) F_{39}\! \left(x \right) F_{8}\! \left(x \right)
F_{42}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{8} \left(x \right)^{3} F_{21}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right) F_{50}\! \left(x \right) F_{8}\! \left(x \right)
F_{47}\! \left(x \right) = F_{37}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = x^{2}
F_{50}\! \left(x \right) = F_{14}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{23}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right) F_{8}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \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_33(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_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_8(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_29(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_24(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_19(x) + F_26(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_28(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_19(x)*F_8(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_36(x) + F_45(x))
Eq(F_36(x), F_14(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x)*F_42(x))
Eq(F_39(x), F_1(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_21(x)*F_39(x)*F_8(x))
Eq(F_42(x), F_1(x) + F_8(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_21(x)*F_8(x)**3)
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_14(x)*F_47(x)*F_50(x)*F_8(x))
Eq(F_47(x), F_37(x) + F_48(x))
Eq(F_48(x), F_40(x) + F_49(x))
Eq(F_49(x), x**2)
Eq(F_50(x), F_14(x) + F_51(x))
Eq(F_51(x), F_19(x) + F_52(x))
Eq(F_52(x), F_23(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_54(x)*F_8(x))
Eq(F_54(x), F_15(x) + F_52(x))
Eq(F_55(x), F_51(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, 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, 2, 3, 1], "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, 0, 3, 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, 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, 2, 3, 1], "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, 0, 3, 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, 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, 2, 3, 1], "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, 0, 3, 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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