0123_0132_0213_0321_1032
Counting sequence:
1, 1, 2, 6, 19, 60, 190, 602, 1907, 6041, 19137, 60623, 192044, 608365, 1927204, 6105077, 19339917, 61265794, 194080332, 614815753, 1947638930, 6169811660, 19544986154, 61915420569, 196138246097, 621334898939, 1968290551801, 6235232727028, 19752229732865, 62571935403258, 198217981112029, 627923170074789, 1989161151298018, 6301347480714854, 19961670801182111, 63235411536067580, 200319768418334430, 634581299376652658, 2010253050400911023, 6368163276503334349, 20173332659844973933, 63905922780960091187, 202443841845486031236, 641310027579644707225, 2031568595641009330116, 6435687547833039961633, 20387238856812923268113, 64583543734141143757346, 204590437702448501865188, 648110103273138312170157, 2053110158430884300757050, 6503927806963408762244180, 20603413189734054409207666, 65268349782790793516966941, 206759794803868313774632133, 654982282984109641380067551, 2074880135328176456764928869, 6572891645809475204327083932, 20821879708592776581966754765, 65960417113450573018050054482, 208952154496642493006039402441, 661927331261303556977724632921, 2096880948302219040387307252962, 6642586736786491534155950509286, 21042662718385439223417261391259, 66659822720499452194048674046780, 211167760686769120610070615905950, 668946020760291748742890419991082, 2119115045003488711393351346675547, 6713020833663504923206146597791761, 21265786781824329146494150017451497, 67366644414719673113526262325714503, 213406859866482540475597728826051484, 676039132329432774796575339915265685, 2141584899035913341470284394379774564, 6784201772425985432248522378465712157, 21491276722070339305863120086401616357, 68080960831953411825733353708049132514, 215669701141676293000226821486699047292, 683207455096743572180158182254434662433, 2164293010232067200886078134093000635970, 6856137472147600776832954691544350869980, 21719157625494613181858953617655222887514, 68802851441851230375168858938897235272049, 217956536259616825478286408065687406920137, 690451786557692102021744176740828315665459, 2187241904931284153633711294482624194652961, 6928835935871234881482537840640350162527108, 21949454844469472023012815467337019711992105, 69532396556713292274284124814235479209616778, 220267619636951062432183738829504513759251589, 697772932663920896461094309582547230539211869, 2210434136260719801908838362908775630324877058, 7002305251499348238316598441571100439283517574, 22182194000188935445439800205048891265169724871, 70269677340424325043675983031381677825619716860, 222603208388010951805752307614378458725101308670, 705171707912911378069819924811662058266359858178, 2233872284419393848851371940292700639185149283383, 7076553592693779125121944453075477157180030750629, 22417400985519149179699720901235377129347710659573
Generating function in Maple syntax:
-(2*x^3+x^2+2*x-1)/(x^4-2*x^3-3*x+1)
Generating function in latex syntax:
-\frac{2 x^{3}+x^{2}+2 x -1}{x^{4}-2 x^{3}-3 x +1}
Generating function in sympy syntax:
(-2*x**3 - x**2 - 2*x + 1)/(x**4 - 2*x**3 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^4-2*x^3-3*x+1)*F(x)+2*x^3+x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-2 x^{3}-3 x +1\right) F \! \left(x \right)+2 x^{3}+x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
358/4595*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-2*_Z^3-3*_Z+1))+93/919*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4-2*_Z^3-3*_Z+1))+104/919*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-2*_Z^3-3*_Z+1))+51/4595*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-2*_Z^3-3*_Z+1))
Explicit closed form in latex syntax:
\frac{358 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{4595}+\frac{93 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{919}+\frac{104 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{919}+\frac{51 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{4595}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+1) = 1/2*a(n)-3/2*a(n+3)+1/2*a(n+4), 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 +1\right) = \frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +3\right)}{2}+\frac{a \! \left(n +4\right)}{2}, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15959/
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[16,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[69,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[28,x]
F[21,x] = F[12,x]+F[22,x]
F[22,x] = F[18,x]+F[23,x]+F[27,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[4,x]
F[26,x] = F[14,x]
F[27,x] = F[11,x]*F[4,x]
F[28,x] = F[29,x]+F[58,x]
F[29,x] = F[18,x]+F[19,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[18,x]+F[33,x]+F[57,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[37,x]
F[35,x] = F[36,x]+F[4,x]
F[36,x] = F[15,x]+F[18,x]+F[23,x]
F[37,x] = F[32,x]+F[38,x]
F[38,x] = F[18,x]+F[39,x]+F[54,x]+F[55,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[43,x]
F[43,x] = F[25,x]*F[4,x]
F[44,x] = F[45,x]
F[45,x] = F[26,x]*F[4,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = 2*F[18,x]+F[39,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[32,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[29,x]*F[4,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[49,x]+F[52,x]
F[57,x] = F[2,x]*F[4,x]
F[58,x] = F[18,x]+F[55,x]+F[59,x]+F[67,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[43,x]
F[63,x] = F[45,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = 2*F[18,x]+F[48,x]+F[59,x]
F[66,x] = F[51,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[29,x]+F[53,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[31,x]+F[68,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_{16}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{69}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{58}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{57}\! \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_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{32}\! \left(x \right)
F_{50}\! \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_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)+F_{67}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{43}\! \left(x \right)
F_{63}\! \left(x \right) = F_{45}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{66}\! \left(x \right) = F_{51}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{29}\! \left(x \right)+F_{53}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{31}\! \left(x \right)+F_{68}\! \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_16(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_69(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_28(x))
Eq(F_21(x), F_12(x) + F_22(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_27(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_4(x))
Eq(F_26(x), F_14(x))
Eq(F_27(x), F_11(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_58(x))
Eq(F_29(x), F_18(x) + F_19(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_57(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_37(x))
Eq(F_35(x), F_36(x) + F_4(x))
Eq(F_36(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_37(x), F_32(x) + F_38(x))
Eq(F_38(x), F_18(x) + F_39(x) + F_54(x) + F_55(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_25(x)*F_4(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_26(x)*F_4(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), 2*F_18(x) + F_39(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_32(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_29(x)*F_4(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_49(x) + F_52(x))
Eq(F_57(x), F_2(x)*F_4(x))
Eq(F_58(x), F_18(x) + F_55(x) + F_59(x) + F_67(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_43(x))
Eq(F_63(x), F_45(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), 2*F_18(x) + F_48(x) + F_59(x))
Eq(F_66(x), F_51(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_29(x) + F_53(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_31(x) + F_68(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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