0123_0231_1032_1203
Counting sequence:
1, 1, 2, 6, 20, 62, 184, 540, 1588, 4684, 13832, 40848, 120608, 356072, 1051216, 3103488, 9162448, 27050416, 79861280, 235775328, 696081920, 2055049760, 6067144576, 17912093760, 52882060096, 156124254016, 460927252352, 1360800301056, 4017504823040, 11860921100672, 35017120217344, 103381406714880, 305213997837568, 901086447126784, 2660286851021312, 7853992424683008, 23187423184575488, 68456469635866112, 202104744356571136, 596675930101337088, 1761572528619947008, 5200708822059314176, 15354106522673285120, 45330087719898451968, 133828487490259189760, 395103229770908149760, 1166467357607739314176, 3443773661767639941120, 10167088651161937186816, 30016401132334483910656, 88617731963440945405952, 261627047950285000237056, 772404243514369061519360, 2280377048447887103762432, 6732380779561524060258304, 19876077507383855949496320, 58680349495213839887515648, 173242603607338758080118784, 511465933022390432789921792, 1510006171665402090513235968, 4458006860777151826828525568, 13161419829706457327001239552, 38856596085092807047297171456, 114716731086430506715867250688, 338679393386318016981939847168, 999886681029148569812887207936, 2951975804914368838850118287360, 8715148744486378123544538513408, 25729823907118970312367335014400, 75962425622417403572617196994560, 224264655959995226965788761522176, 662098866653521990444510005166080, 1954721341833179801684854038593536, 5770944063883456506038188050743296, 17037617933428141252233933581975552, 50300335895152142569041710212448256, 148502202658329277882677190700564480, 438424591047331736771086165217902592, 1294365461213170403196187185658200064, 3821368557770317506603841883587215360, 11281865973639910442938145503552995328, 33307569767999376025271761181615325184, 98334283215404838148934520850740150272, 290313322852442304393735013450779721728, 857095030031428437915621783770494926848, 2530410534682752962435959645343571771392, 7470557230740992383166690514895347122176, 22055403490000699167541637072649650700288, 65114396166467189938202949089245347708928, 192237906236713453773850548386860625821696, 567545961722468438444007052365551108096000, 1675571821255955716892115188238551925915648, 4946808042940310401216539538444215218339840, 14604512622655782521733412602327476637532160, 43117053884818736792739736736741251744792576, 127294924777044349320242605991020634717552640, 375814124900094635955911381298660915602784256, 1109519933507150185171621399908698580305575936, 3275647191751948369088804759895674431735857152, 9670727132333470706221041414527266506361798656, 28550987879140578990784012180941087290320486400
Generating function in Maple syntax:
-(2*x-1)*(x-1)/(2*x^4+2*x^3-4*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)}{2 x^{4}+2 x^{3}-4 x^{2}+4 x -1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)/(2*x**4 + 2*x**3 - 4*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4+2*x^3-4*x^2+4*x-1)*F(x)+(2*x-1)*(x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}+2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right) = 0
Explicit closed form in Maple syntax:
-23/571*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))-30/571*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+161/571*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+3/571*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))
Explicit closed form in latex syntax:
-\frac{23 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{571}-\frac{30 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{571}+\frac{161 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{571}+\frac{3 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{571}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = 2*a(n)+2*a(n+1)-4*a(2+n)+4*a(n+3), n >= 4
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(n +4\right) = 2 a \! \left(n \right)+2 a \! \left(n +1\right)-4 a \! \left(2+n \right)+4 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14978/
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[19,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[18,x]
F[17,x] = F[4,x]
F[18,x] = F[15,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[68,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[37,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[4,x]
F[28,x] = F[21,x]+F[29,x]+F[35,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[32,x]+F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]*F[4,x]
F[34,x] = F[15,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[7,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[21,x]+F[22,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[21,x]+F[44,x]+F[53,x]+F[66,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[4,x]
F[49,x] = F[50,x]
F[50,x] = 2*F[21,x]+F[44,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[41,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[61,x]
F[55,x] = F[41,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[2,x]+F[58,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[41,x]
F[65,x] = F[62,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[20,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[60,x]+F[70,x]
F[70,x] = F[20,x]+F[71,x]
F[71,x] = F[62,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_{19}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{68}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{53}\! \left(x \right)+F_{66}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \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_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{44}\! \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_{41}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \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_{2}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{41}\! \left(x \right)
F_{65}\! \left(x \right) = F_{62}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{20}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{60}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{20}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{62}\! \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_19(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) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_15(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_68(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_37(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_4(x))
Eq(F_28(x), F_21(x) + F_29(x) + F_35(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_32(x) + F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x)*F_4(x))
Eq(F_34(x), F_15(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_7(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_21(x) + F_22(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_21(x) + F_44(x) + F_53(x) + F_66(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_4(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), 2*F_21(x) + F_44(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_41(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_61(x))
Eq(F_55(x), F_41(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_2(x) + F_58(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_41(x))
Eq(F_65(x), F_62(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_20(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_60(x) + F_70(x))
Eq(F_70(x), F_20(x) + F_71(x))
Eq(F_71(x), F_62(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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