0132_0231_0321_1023_2103
Counting sequence:
1, 1, 2, 6, 19, 52, 139, 377, 1033, 2828, 7726, 21099, 57638, 157479, 430255, 1175475, 3211434, 8773790, 23970467, 65488568, 178918079, 488813221, 1335462537, 3648551580, 9968028370, 27233159899, 74402376338, 203271072339, 555346897555, 1517235940123, 4145165675290, 11324803230290, 30939937810891, 84529482082964, 230938839788515, 630936643742625, 1723750967060873, 4709375221606524, 12866252377336534, 35151255197887995, 96035015150447790, 262372540696667951, 716815111694230871, 1958375304781802531, 5350380832952071034, 14617512275467742854, 39935786216839618659, 109106596984614723072, 298084766402908696855, 814382726775046848925, 2224934986355911078121, 6078635426261915831628, 16607140825235653823866, 45371552502995139346891, 123957386656461586359610, 338657878318913451372731, 925230529950750075410683, 2527776816539327053589003, 6906014692980154258093642, 18867583019038962623397114, 51547195424038233762865067, 140829556886154392772398268, 384753504620385253070611291, 1051166123013079291686261657, 2871839255266929089513787369, 7846010756560016762399770892, 21435700023653891703826832510, 58563421560427816932453492491, 159998243168163417272561261174, 437123329457182468410029505655, 1194243145250691771365180636799, 3262732949415748479550419675411, 8913952189332880501831201522954, 24353370277497257962763243903086, 66534644933660276929188890563043, 181776030422315069783904266527368, 496621350711950693426186312963503, 1356794762268531526420181161675669, 3706832225960964439692734952900553, 10127253976458991932225832227675836, 27668172404839912743837134354836642, 75590852762597809352125933162879355, 206518050334875444191926135043224738, 564217806194946507088104136420669923, 1541471713059643902560060542922142179, 4211379038509180819296329358669369723, 11505701503137649443712779803180209210, 31434161083293660526018218323721059490, 85879725172862619939461996253821606475, 234627772512312560930960429155066244900, 641014995370350361740844850817734732051, 1751285535765325845343610559945601971857, 4784601062271352414168910821526733465545, 13071773196073356519025042762944711827548, 35712748516689417866387907168942830510502, 97569043425525548770825899863774983665627, 266563583884429933274427614065435647475198, 728265254619910964090507027858421423367807, 1989657677008681794729869283847714223573543, 5435845863257185517640752623412271113673603, 14851007080531734624741243814519970431520602
Generating function in Maple syntax:
(x^2+x-1)/(2*x^2+2*x-1)/(x^3+x^2+1)
Generating function in latex syntax:
\frac{x^{2}+x -1}{\left(2 x^{2}+2 x -1\right) \left(x^{3}+x^{2}+1\right)}
Generating function in sympy syntax:
(x**2 + x - 1)/((2*x**2 + 2*x - 1)*(x**3 + x**2 + 1))
Implicit equation for the generating function in Maple syntax:
(2*x^2+2*x-1)*(x^3+x^2+1)*F(x)-x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}+2 x -1\right) \left(x^{3}+x^{2}+1\right) F \! \left(x \right)-x^{2}-x +1 = 0
Explicit closed form in Maple syntax:
110/93*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^5+4*_Z^4+_Z^3+_Z^2+2*_Z-1))+64/93*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^5+4*_Z^4+_Z^3+_Z^2+2*_Z-1))-29/93*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^5+4*_Z^4+_Z^3+_Z^2+2*_Z-1))+42/31*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^5+4*_Z^4+_Z^3+_Z^2+2*_Z-1))-12/31*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^5+4*_Z^4+_Z^3+_Z^2+2*_Z-1))
Explicit closed form in latex syntax:
\frac{110 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{93}+\frac{64 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{93}-\frac{29 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{93}+\frac{42 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{31}-\frac{12 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{31}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = 2*a(n)+4*a(n+1)+a(n+2)+a(n+3)+2*a(n+4), n >= 5
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(n +5\right) = 2 a \! \left(n \right)+4 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18641/
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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[4,x]
F[17,x] = F[15,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[20,x]+F[21,x]+F[57,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[1,x]+F[4,x]
F[27,x] = F[20,x]+F[28,x]+F[30,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[16,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[11,x]+F[17,x]
F[32,x] = F[33,x]+F[38,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[2,x]*F[4,x]
F[38,x] = 2*F[20,x]+F[39,x]+F[44,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[36,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[42,x]+F[46,x]
F[46,x] = F[20,x]+F[47,x]+F[55,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[4,x]+F[50,x]
F[50,x] = F[20,x]+F[28,x]+F[51,x]
F[51,x] = F[11,x]*F[4,x]
F[52,x] = F[36,x]+F[53,x]
F[53,x] = 2*F[20,x]+F[39,x]+F[54,x]
F[54,x] = F[4,x]*F[46,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[2,x]+F[46,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[56,x]+F[59,x]
F[59,x] = F[46,x]+F[60,x]
F[60,x] = F[43,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_{11}\! \left(x \right)+F_{14}\! \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_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{44}\! \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_{41}\! \left(x \right) = F_{36}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{20}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \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_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{36}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{46}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{43}\! \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_11(x) + F_14(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_4(x))
Eq(F_17(x), F_15(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_57(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_1(x) + F_4(x))
Eq(F_27(x), F_20(x) + F_28(x) + F_30(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_16(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_11(x) + F_17(x))
Eq(F_32(x), F_33(x) + F_38(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_2(x)*F_4(x))
Eq(F_38(x), 2*F_20(x) + F_39(x) + F_44(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_36(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_42(x) + F_46(x))
Eq(F_46(x), F_20(x) + F_47(x) + F_55(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_4(x) + F_50(x))
Eq(F_50(x), F_20(x) + F_28(x) + F_51(x))
Eq(F_51(x), F_11(x)*F_4(x))
Eq(F_52(x), F_36(x) + F_53(x))
Eq(F_53(x), 2*F_20(x) + F_39(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_46(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_2(x) + F_46(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_56(x) + F_59(x))
Eq(F_59(x), F_46(x) + F_60(x))
Eq(F_60(x), F_43(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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