0123_0231_2031
Counting sequence:
1, 1, 2, 6, 21, 76, 274, 978, 3463, 12201, 42869, 150415, 527426, 1848905, 6480722, 22715293, 79617891, 279063942, 978133274, 3428414441, 12016810218, 42119714480, 147632440224, 517461749617, 1813738775801, 6357278456431, 22282695956167, 78102374002636, 273754164782411, 959527077451746, 3363208053812049, 11788274322402253, 41318707985800198, 144824898263178786, 507621176421643729, 1779246951574641540, 6236382290003293526, 21858946579942265890, 76617103212323215187, 268548188412508436813, 941279772739569456581, 3299249999809549422007, 11564096962971883118574, 40532950996966838159601, 142070766250332720758378, 497967755283037350149045, 1745411050044558003787959, 6117785140297066000269482, 21443255456578058681090526, 75160077386068054765906261, 263441213211252694456304282, 923379475272881608474088716, 3236508232565346738697168576, 11344182776390654993385301741, 39762136727875621569810744149, 139369009502975080610848475103, 488497913046588748850270137983, 1712218605139606689175509877096, 6001443350089851106466693350939, 21035469522538567960529931929490, 73730759789147085422148757756801, 258431357249251606009569570707625, 905819587383678273440625837804602, 3174959624178165709959760546116958, 11128450693230388822535838134581421, 39005981017385811408534751221340172, 136718632006357743818284618528302074, 479208158599022170577058762402460786, 1679657380255141486198736443059410463, 5887314029238486653388595308708709185, 20635438444954379951164352890059941669, 72328623494639787498984079206704608847, 253516773611781821963103033198716892426, 888593635509882013114111119618457561241, 3114581484370758825521413356373417348722, 10916821182518225713810605807880523128621, 38264205103998441506039873528251239500891, 134118656682358456639266875402042776646686, 470095067214677015895014392603486676267074, 1647715371493427308979957153763598091149153, 5775355103260181976971086429733244972321818, 20243014749884839747730691418157830485665256, 70953151596294917880611814026252807089794624, 248695650507069254951246974048218997196976265, 871695269197373309106256214482497958824468001, 3055351554366979415609215238719047564419784735, 10709216225720967779240029433909673149235288967, 37536535527417122367566608298473265445882090228, 131568125033929963996336031845377259758813590411, 461155279295828400755819256500646852573616380562, 1616380803234596191157570713855388508668940613649, 5665525297801687054855891513917231759790466201621, 19858053767898078044062570671207330336584402149598, 69603837017856414904682262700249102763087675270650, 243966210598145485822284477215701513067322997482729, 855118258758483793733383452161075876694668409538212, 2997247998685354357510496495466172375458985154384126, 10505559287982207359637276126605066475751460467023746, 36822704027734118452093503857384665057439849540106987, 129066096791743021034404909556054777416397366241132101, 452385499134150091969728681460301214529887462121392981
Generating function in Maple syntax:
-(x-1)*(3*x-1)*(2*x-1)/(3*x^4-14*x^3+16*x^2-7*x+1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right)}{3 x^{4}-14 x^{3}+16 x^{2}-7 x +1}
Generating function in sympy syntax:
(1 - x)*(2*x - 1)*(3*x - 1)/(3*x**4 - 14*x**3 + 16*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(3*x^4-14*x^3+16*x^2-7*x+1)*F(x)+(x-1)*(3*x-1)*(2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{4}-14 x^{3}+16 x^{2}-7 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) = 0
Explicit closed form in Maple syntax:
450/491*Sum(_alpha^(-n+2),_alpha = RootOf(3*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1))-1755/491*Sum(_alpha^(-n+1),_alpha = RootOf(3*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1))+1300/491*Sum(_alpha^(-n),_alpha = RootOf(3*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1))-217/491*Sum(_alpha^(-n-1),_alpha = RootOf(3*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1))
Explicit closed form in latex syntax:
\frac{450 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{491}-\frac{1755 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{491}+\frac{1300 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{491}-\frac{217 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{491}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -3*a(n)+14*a(n+1)-16*a(n+2)+7*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) = -3 a \! \left(n \right)+14 a \! \left(n +1\right)-16 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14641/
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[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[48,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = 2*F[21,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[25,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[47,x]
F[34,x] = F[35,x]
F[35,x] = F[21,x]+F[36,x]+F[45,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[11,x]+F[14,x]
F[39,x] = F[35,x]+F[40,x]
F[40,x] = 2*F[21,x]+F[32,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[14,x]
F[44,x] = F[40,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[2,x]+F[35,x]
F[47,x] = F[40,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[46,x]+F[50,x]
F[50,x] = F[20,x]+F[40,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_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \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_{48}\! \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_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{45}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \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_{14}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{47}\! \left(x \right) = F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \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_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_48(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), 2*F_21(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_47(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_21(x) + F_36(x) + F_45(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_11(x) + F_14(x))
Eq(F_39(x), F_35(x) + F_40(x))
Eq(F_40(x), 2*F_21(x) + F_32(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_14(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_2(x) + F_35(x))
Eq(F_47(x), F_40(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_46(x) + F_50(x))
Eq(F_50(x), F_20(x) + F_40(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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