0132_0213_1302_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 55, 148, 383, 976, 2481, 6327, 16214, 41748, 107918, 279828, 727288, 1893599, 4936805, 12883640, 33648069, 87928868, 229875631, 601172827, 1572593196, 4114508344, 10766736060, 28177309528, 73748413358, 193034373891, 505287596931, 1322694196332, 3462526553419, 9064348589448, 23729445469133, 62121840329903, 162631780548418, 425764911375404, 1114645773702730, 2918138049987980, 7639699656777444, 20000823481383223, 52362495909456961, 137086114491164752, 358894748052399761, 939595930642768444, 2459888645829383115, 6440061210752346483, 16860277394241598808, 44140735787600347056, 115561859599815249656, 302544702274357030512, 792071965748279114074, 2073670632020526872123, 5428938804413394640223, 14213143529419843342628, 37210487280246507995223, 97418309304120425878784, 255044422617716260134505, 667714922520231335534695, 1748100272885383708514094, 4576585752020731714122628, 11981656694946435282111366, 31368383756357821828755652, 82123493421205525597274576, 215002094201415745749338319, 562882784571355693223314845, 1473646250289279297065790888, 3858055947849738124264464733, 10100521556366446928308456596, 26443508647462626365822653031, 69230004238447479579483041867, 181246503772731907193273596804, 474509506489452431641632044648, 1242282014515033767014211179188, 3252336534694465627966178829128, 8514727584845996634014680035078, 22291846210398791308338570786739, 58360811027460911359522451405755, 152790586834205010907271621653660, 400010949399596257636378090066019, 1047242261213468034550034001633208, 2741715833938576391110066621081221, 7177905239997798228972851274178527, 18791999884845892476193858026666266, 49198094412122027560379464456322844, 128802283346684486926486018643389250, 337208755618260026662161558076103868, 882823983488752779946164588789528508, 2311263194809312686948664074701785511, 6050965600861814028444491368134530681, 15841633607621386893474137495339310272, 41473935221692861642156576049158509609, 108580172056838228013352900514686543148, 284266580947583882358616745220001878403, 744219570783437538983926574595520722403, 1948392131397776974436021457466963666608, 5100956823399989864009854755606177154144, 13354478338782385576964976724953181674480, 34962478192907552785627943250456595756000, 91532956239861044617404588688823061500594, 239636390526517124741557294140825500697963
Generating function in Maple syntax:
(x^7-8*x^6+20*x^5-38*x^4+43*x^3-26*x^2+8*x-1)/(x^2-3*x+1)/(2*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{7}-8 x^{6}+20 x^{5}-38 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 - 8*x**6 + 20*x**5 - 38*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)*(x-1)^4*F(x)-x^7+8*x^6-20*x^5+38*x^4-43*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{7}+8 x^{6}-20 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/6*n^3-3/2*n^2-3+7/3*n+2^(-1+n)+(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{n^{3}}{6}-\frac{3 n^{2}}{2}-3+\frac{7 n}{3}+2^{-1+n}+\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-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) = 55
a(6) = 148
a(7) = 383
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1/6*(n+1)*(n^2-10*n+12), 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) = 55
a \! \left(6\right) = 148
a \! \left(7\right) = 383
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{\left(n +1\right) \left(n^{2}-10 n +12\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18525/
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[16,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[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[27,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[20,x]+F[5,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[17,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[28,x]+F[29,x]+F[54,x]
F[28,x] = 0
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[38,x]
F[31,x] = F[32,x]+F[6,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[27,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]+F[52,x]
F[43,x] = F[28,x]+F[44,x]+F[50,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[33,x]
F[48,x] = F[43,x]+F[49,x]
F[49,x] = F[40,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[17,x]+F[43,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[42,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[51,x]+F[56,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[31,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[56,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_{16}\! \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_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \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_{24}\! \left(x \right) = F_{17}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{54}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
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_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{43}\! \left(x \right) = F_{28}\! \left(x \right)+F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{33}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right)+F_{43}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{31}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \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_16(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_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_27(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_5(x))
Eq(F_20(x), F_17(x) + F_21(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))
Eq(F_24(x), F_17(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_54(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_38(x))
Eq(F_31(x), F_32(x) + F_6(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_27(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x) + F_52(x))
Eq(F_43(x), F_28(x) + F_44(x) + F_50(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_33(x))
Eq(F_48(x), F_43(x) + F_49(x))
Eq(F_49(x), F_40(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_17(x) + F_43(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_42(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_51(x) + F_56(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_31(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_56(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:
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