0132_0213_1203_1302_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 137, 317, 683, 1394, 2734, 5216, 9783, 18202, 33843, 63227, 119111, 226646, 435680, 845350, 1653251, 3253758, 6434653, 12770361, 25408403, 50642282, 101057234, 201821708, 403270303, 806069618, 1611549927, 3222368503, 6443836239, 12886570862, 25771803332, 51541990610, 103082041163, 206161765942, 412320780305, 824638307829, 1649272787899, 3298541090850, 6597076948214, 13194147813176, 26388288581447, 52776569032970, 105553128715291, 211106246710259, 422212481167383, 844424948370502, 1688849880871080, 3377699743754750, 6755399467174355, 13510798911415982, 27021597797030981, 54043195565099953, 108086391097760675, 216172782159263898, 432345564278084890, 864691128511146468, 1729382256972265007, 3458764513889042402, 6917529027716649935, 13835058055365395951, 27670116110655861151, 55340232221229169118, 110680464442367527308, 221360928884635308970, 442721857769161216923, 885443715538202611046, 1770887431076274163193, 3541774862152405166957, 7083549724304654157131, 14167099448609138148562, 28334198897218091113790, 56668397794435980938256, 113336795588871743330647, 226673591177743249643514, 453347182355486242514435, 906694364710972207148267, 1813388729421944113881575, 3626777458843887903311414, 7253554917687775456552816, 14507109835375550535753686, 29014219670751100665124515, 58028439341502200892997726, 116056878683004401315946285, 232113757366008802127020841, 464227514732017603712223923, 928455029464035206843458250, 1856910058928070413064423266, 3713820117856140825462408124, 7427640235712281650211877567, 14855280471424563299661643602, 29710560942849126598509208759, 59421121885698253196149452519, 118842243771396506391372004079, 237684487542793012781755987790, 475368975085586025562459513940, 950737950171172051123798660226, 1901475900342344102246405434603
Generating function in Maple syntax:
(x^8-7*x^7+22*x^6-47*x^5+63*x^4-54*x^3+28*x^2-8*x+1)/(2*x-1)/(x-1)^7
Generating function in latex syntax:
\frac{x^{8}-7 x^{7}+22 x^{6}-47 x^{5}+63 x^{4}-54 x^{3}+28 x^{2}-8 x +1}{\left(2 x -1\right) \left(x -1\right)^{7}}
Generating function in sympy syntax:
(x**8 - 7*x**7 + 22*x**6 - 47*x**5 + 63*x**4 - 54*x**3 + 28*x**2 - 8*x + 1)/((x - 1)**7*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^7*F(x)-x^8+7*x^7-22*x^6+47*x^5-63*x^4+54*x^3-28*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{7} F \! \left(x \right)-x^{8}+7 x^{7}-22 x^{6}+47 x^{5}-63 x^{4}+54 x^{3}-28 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1+3*2^(-1+n)-1/16*n^3-193/360*n^2-13/30*n+5/144*n^4-1/240*n^5+1/720*n^6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -1+3 \,2^{-1+n}-\frac{n^{3}}{16}-\frac{193 n^{2}}{360}-\frac{13 n}{30}+\frac{5 n^{4}}{144}-\frac{n^{5}}{240}+\frac{n^{6}}{720} & \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) = 54
a(6) = 137
a(7) = 317
a(8) = 683
a(n+1) = 2*a(n)-1/720*n*(-1+n)*(n^4-8*n^3+17*n^2-118*n-504), n >= 9
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) = 54
a \! \left(6\right) = 137
a \! \left(7\right) = 317
a \! \left(8\right) = 683
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{n \left(-1+n \right) \left(n^{4}-8 n^{3}+17 n^{2}-118 n -504\right)}{720}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18389/
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[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[31,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[1,x]+F[14,x]
F[25,x] = F[21,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[14,x]
F[30,x] = F[26,x]
F[31,x] = F[32,x]+F[33,x]+F[37,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[6,x]
F[36,x] = F[31,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[46,x]
F[39,x] = F[21,x]+F[40,x]
F[40,x] = F[32,x]+F[41,x]+F[45,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[10,x]
F[44,x] = F[40,x]
F[45,x] = F[12,x]*F[39,x]
F[46,x] = F[47,x]+F[56,x]
F[47,x] = F[27,x]+F[32,x]+F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[55,x]
F[50,x] = F[21,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[43,x]+F[54,x]
F[54,x] = F[51,x]
F[55,x] = F[47,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[46,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{39}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{47}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{21}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{43}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{47}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_20(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_31(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_1(x) + F_14(x))
Eq(F_25(x), F_21(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_14(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_37(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_6(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_46(x))
Eq(F_39(x), F_21(x) + F_40(x))
Eq(F_40(x), F_32(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_10(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_12(x)*F_39(x))
Eq(F_46(x), F_47(x) + F_56(x))
Eq(F_47(x), F_27(x) + F_32(x) + F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_55(x))
Eq(F_50(x), F_21(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_43(x) + F_54(x))
Eq(F_54(x), F_51(x))
Eq(F_55(x), F_47(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_46(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, 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": [1, 2, 0, 3], "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, 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": [1, 2, 0, 3], "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, 1, 2, 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