0123_0231_2103_2130_3012
Counting sequence:
1, 1, 2, 6, 19, 48, 100, 200, 396, 784, 1556, 3096, 6172, 12320, 24612, 49192, 98348, 196656, 393268, 786488, 1572924, 3145792, 6291524, 12582984, 25165900, 50331728, 100663380, 201326680, 402653276, 805306464, 1610612836, 3221225576, 6442451052, 12884902000, 25769803892, 51539607672, 103079215228, 206158430336, 412316860548, 824633720968, 1649267441804, 3298534883472, 6597069766804, 13194139533464, 26388279066780, 52776558133408, 105553116266660, 211106232533160, 422212465066156, 844424930132144, 1688849860264116, 3377699720528056, 6755399441055932, 13510798882111680, 27021597764223172, 54043195528446152, 108086391056892108, 216172782113784016, 432345564227567828, 864691128455135448, 1729382256910270684, 3458764513820541152, 6917529027641082084, 13835058055282163944, 27670116110564327660, 55340232221128655088, 110680464442257309940, 221360928884514619640, 442721857769029239036, 885443715538058477824, 1770887431076116955396, 3541774862152233910536, 7083549724304467820812, 14167099448608935641360, 28334198897217871282452, 56668397794435742564632, 113336795588871485128988, 226673591177742970257696, 453347182355485940515108, 906694364710971881029928, 1813388729421943762059564, 3626777458843887524118832, 7253554917687775048237364, 14507109835375550096474424, 29014219670751100192948540, 58028439341502200385896768, 116056878683004400771793220, 232113757366008801543586120, 464227514732017603087171916, 928455029464035206174343504, 1856910058928070412348686676, 3713820117856140824697373016, 7427640235712281649394745692, 14855280471424563298789491040, 29710560942849126597578981732, 59421121885698253195157963112, 118842243771396506390315925868, 237684487542793012780631851376, 475368975085586025561263702388, 950737950171172051122527404408, 1901475900342344102245054808444
Generating function in Maple syntax:
-(x+1)*(2*x^6-11*x^5+9*x^4-6*x^3+7*x^2-4*x+1)/(x-1)^2/(2*x-1)
Generating function in latex syntax:
-\frac{\left(x +1\right) \left(2 x^{6}-11 x^{5}+9 x^{4}-6 x^{3}+7 x^{2}-4 x +1\right)}{\left(x -1\right)^{2} \left(2 x -1\right)}
Generating function in sympy syntax:
(-x - 1)*(2*x**6 - 11*x**5 + 9*x**4 - 6*x**3 + 7*x**2 - 4*x + 1)/((x - 1)**2*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)*F(x)+(x+1)*(2*x^6-11*x^5+9*x^4-6*x^3+7*x^2-4*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right) F \! \left(x \right)+\left(x +1\right) \left(2 x^{6}-11 x^{5}+9 x^{4}-6 x^{3}+7 x^{2}-4 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,19,-20+3*2^(-1+n)+4*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ -20+3 \,2^{-1+n}+4 n & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 48
a(6) = 100
a(7) = 200
a(n+1) = 2*a(n)-4*n+24, n >= 8
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) = 48
a \! \left(6\right) = 100
a \! \left(7\right) = 200
a \! \left(n +1\right) = 2 a \! \left(n \right)-4 n +24, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17619/
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[18,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[17,x] = F[11,x]
F[18,x] = F[19,x]+F[32,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[31,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[11,x]+F[28,x]
F[31,x] = F[22,x]*F[4,x]
F[32,x] = F[24,x]+F[33,x]+F[43,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]+F[7,x]
F[36,x] = F[24,x]+F[37,x]+F[38,x]
F[37,x] = F[17,x]*F[4,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[11,x]+F[41,x]
F[41,x] = F[37,x]
F[42,x] = F[36,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[53,x]
F[45,x] = F[19,x]+F[46,x]
F[46,x] = F[24,x]+F[31,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[11,x]+F[50,x]
F[50,x] = F[24,x]+F[37,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[11,x]+F[50,x]
F[53,x] = F[54,x]+F[58,x]
F[54,x] = F[24,x]+F[25,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[22,x]+F[57,x]
F[57,x] = F[54,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[41,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_{18}\! \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_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{32}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{24}\! \left(x \right)+F_{33}\! \left(x \right)+F_{43}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{22}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{41}\! \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_18(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))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_32(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_31(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_11(x) + F_28(x))
Eq(F_31(x), F_22(x)*F_4(x))
Eq(F_32(x), F_24(x) + F_33(x) + F_43(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x) + F_7(x))
Eq(F_36(x), F_24(x) + F_37(x) + F_38(x))
Eq(F_37(x), F_17(x)*F_4(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_11(x) + F_41(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_36(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_53(x))
Eq(F_45(x), F_19(x) + F_46(x))
Eq(F_46(x), F_24(x) + F_31(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_11(x) + F_50(x))
Eq(F_50(x), F_24(x) + F_37(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_11(x) + F_50(x))
Eq(F_53(x), F_54(x) + F_58(x))
Eq(F_54(x), F_24(x) + F_25(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_22(x) + F_57(x))
Eq(F_57(x), F_54(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_41(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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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