0231_0312_0321_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 57, 164, 465, 1318, 3747, 10673, 30417, 86677, 246954, 703544, 2004288, 5709961, 16267098, 46343468, 132028305, 376136329, 1071576858, 3052820825, 8697197112, 24777490993, 70588726923, 201100603307, 572916587865, 1632185138134, 4649940985152, 13247241785858, 37740138104553, 107518081674334, 306308839020438, 872644892846543, 2486082711292413, 7082614357826920, 20177697996045885, 57484351942707902, 163767478278352227, 466558046412103945, 1329179718465852429, 3786707222321469869, 10787970496669630386, 30733906955094806332, 87557992211419730940, 249444433188933738425, 710643582355159727430, 2024556309741608105956, 5767769319368986445349, 16431828920431474300277, 46812725461057038707958, 133365024411096068393941, 379944332678694552380256, 1082425445292745397554049, 3083727651250470565212987, 8785248229742883428197831, 25028340757298480506634193, 71303374097351095071620582, 203136564543704753383782348, 578716847231941764039874750, 1648709527122194579215892769, 4697017406396769299302716170, 13381358057961354904538515398, 38122222675076362292278713931, 108606604456223382587911188609, 309409937401739743984976562620, 881479628631025872594155756169, 2511252037398596983901464104598, 7154319385840694336977943522163, 20381978834604634516855003114289, 58066328718347435321192169741609, 165425475033007688506031002345237, 471281522250767730267144164095290, 1342636454093201505916209826084688, 3825044188557785097065870159527656, 10897188885208125160383878504902873, 31045059807446864949989675699871426, 88444437239792348773237688417655932, 251969831180263415744573539854623865, 717838202224955508316484716962451777, 2045053100840901313910851423948581858, 5826162737363423684441753433052111985, 16598186241855835237673813372940443880, 47286661725485414051788545334389264769, 134715223974404442882633235550693141115, 383790923449622221614377834537312285219, 1093384018351924034691788381399687172825, 3114947588760067937347641059003360485846, 8874190877005411160885688251604084960648, 25281729941682163593687110655414521927690, 72025255902522849772628695818444169872489, 205193137486648359688694952080732758197382, 584575828909200436740609200459381130109206, 1665401211417777648761347021299281223302855, 4744570435228525757089532322895665640589445, 13516832136612137949119435263087117073847712, 38508175503679827727102786383443230346663541, 109706147537752617815306977868217962401761326, 312542431578588875573204798170873042216007187, 890403808076861272606052373383324900825190121
Generating function in Maple syntax:
(x^2+x-1)*(x-1)^2/(3*x^4+x^3-4*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{2}}{3 x^{4}+x^{3}-4 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**2*(x**2 + x - 1)/(3*x**4 + x**3 - 4*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(3*x^4+x^3-4*x^2+4*x-1)*F(x)-(x^2+x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{4}+x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-645/11419*Sum(_alpha^(-n+2),_alpha = RootOf(3*_Z^4+_Z^3-4*_Z^2+4*_Z-1))-229/11419*Sum(_alpha^(-n+1),_alpha = RootOf(3*_Z^4+_Z^3-4*_Z^2+4*_Z-1))+86/601*Sum(_alpha^(-n),_alpha = RootOf(3*_Z^4+_Z^3-4*_Z^2+4*_Z-1))+698/11419*Sum(_alpha^(-n-1),_alpha = RootOf(3*_Z^4+_Z^3-4*_Z^2+4*_Z-1))+piecewise(n = 0,1/3,0)
Explicit closed form in latex syntax:
-\frac{645 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{11419}-\frac{229 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{11419}+\frac{86 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{601}+\frac{698 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{11419}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+4) = 3*a(n)+a(n+1)-4*a(n+2)+4*a(n+3), 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 +4\right) = 3 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20623/
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[13,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[6,x]
F[10,x] = F[11,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[4,x]*F[7,x]
F[13,x] = F[14,x]+F[2,x]
F[14,x] = F[15,x]+F[16,x]+F[39,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]+F[7,x]
F[19,x] = F[15,x]+F[20,x]+F[22,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[18,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[10,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[2,x]+F[25,x]
F[29,x] = F[30,x]+F[32,x]
F[30,x] = F[31,x]
F[31,x] = F[2,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]*F[4,x]
F[34,x] = 2*F[15,x]+F[35,x]+F[37,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[24,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[29,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[13,x]+F[41,x]
F[41,x] = F[30,x]+F[42,x]
F[42,x] = F[33,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_{13}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{2}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{39}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \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_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{15}\! \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_{24}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{29}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \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_13(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_6(x))
Eq(F_10(x), F_11(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_4(x)*F_7(x))
Eq(F_13(x), F_14(x) + F_2(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_39(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x) + F_7(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_22(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_18(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_10(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_2(x) + F_25(x))
Eq(F_29(x), F_30(x) + F_32(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_2(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_25(x)*F_4(x))
Eq(F_34(x), 2*F_15(x) + F_35(x) + F_37(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_24(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_29(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_13(x) + F_41(x))
Eq(F_41(x), F_30(x) + F_42(x))
Eq(F_42(x), F_33(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 3, 0, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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