0231_0312_0321_1203
Counting sequence:
1, 1, 2, 6, 20, 66, 214, 686, 2186, 6946, 22042, 69906, 221650, 702706, 2227714, 7062146, 22387746, 70971266, 224985282, 713223106, 2260979330, 7167500546, 22721597314, 72029429506, 228339522306, 723856035586, 2294686238722, 7274353839106, 23060330807810, 73103243075586, 231743603008514, 734647264267266, 2328895364900866, 7382799725051906, 23404113641871362, 74193064387579906, 235198430816473090, 745599367206989826, 2363614478419853314, 7492862317632487426, 23753021579276263426, 75299132725000372226, 238704763106183987202, 756714743816182677506, 2398851183602361991170, 7604565720564714635266, 24107131027421769203714, 76421690301352923561986, 242263367709438726569986, 767995828180205520748546, 2434613196699238782730242, 7717934395017666636283906, 24466519530311777938178050, 77560982937813300914683906, 245875023900594314509811714, 779445090666127100333522946, 2470908348987693170485100546, 7832993166817329852253732866, 24831265787961678867377684482, 78717260120236669424033071106, 249540522571388660256869449730, 791065038469157944444656287746, 2507744588493381118416820109314, 7949767231892431264264228962306, 25201449673644646931333963055106, 79890775053806962959973754077186, 253260666404153695448698632273922, 802858216161711562012179094831106, 2545129981730924931231371606622210, 8068282161791969210003730972278786, 25577152251382603853527822486208514, 81081784718484627131641084787032066, 257036270047589362399501232231153666, 814827206250622987627667114378133506, 2583072715470343971448383823476162562, 8188563909284951185739198600686075906, 25958455793697870824804692527035711490, 82290549925280892984843605259818369026, 260868160295157927918286638783829901314, 826974629742674184053849959561611444226, 2621581098529819094409139350159149236226, 8310638814043612153692284203945714253826, 26345443799629462057843158196341335654402, 83517335373370985735169785677155615637506, 264757176266137794517608257908048362733570, 839303146718550282484407143885199548350466, 2660663563595183569432072488370390236659714, 8434533608411358123724259149895065124470786, 26738201013017964791064077335069080927862786, 84762409708058775777943646063745412989190146, 268704169589376457282197911285053359852093442, 851815456915352347990776198342584451574267906, 2700328669066538925842821678672740260614504450, 8560275423256698092825338495435685419605295106, 27136813441063009823634158227712092254043635714, 86026045579605565257200312562631517655734943746, 272710004589782845195882256804230515275641913346, 864514300317794230621860085636234841283010494466, 2740585100932400101581275203811215299269288263682, 8687891793914446254689616130363930483959848239106, 27541368373157396354352193942560821654734795440130
Generating function in Maple syntax:
-(2*x-1)*(x^2+2*x-1)/(x-1)/(2*x^3+2*x^2-4*x+1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{2}+2 x -1\right)}{\left(x -1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right)}
Generating function in sympy syntax:
(1 - 2*x)*(x**2 + 2*x - 1)/((x - 1)*(2*x**3 + 2*x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x^3+2*x^2-4*x+1)*F(x)+(2*x-1)*(x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
270/37*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-6*_Z^2+5*_Z-1))+239/37*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-6*_Z^2+5*_Z-1))-16*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-6*_Z^2+5*_Z-1))+157/37*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-6*_Z^2+5*_Z-1))
Explicit closed form in latex syntax:
\frac{270 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{37}+\frac{239 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{37}-16 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)+\frac{157 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-6 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{37}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = -2*a(n)-2*a(n+1)+4*a(n+2)+2, 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 +3\right) = -2 a \! \left(n \right)-2 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15769/
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[10,x]*F[4,x]
F[4,x] = F[13,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[6,x]
F[13,x] = F[14,x]+F[2,x]
F[14,x] = F[15,x]+F[16,x]+F[44,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[19,x]+F[6,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[10,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = 2*F[15,x]+F[24,x]+F[28,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[19,x]
F[27,x] = F[23,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[15,x]+F[31,x]+F[43,x]
F[31,x] = F[10,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[10,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = x^2
F[36,x] = F[30,x]+F[37,x]
F[37,x] = 2*F[15,x]+F[38,x]+F[42,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[34,x]
F[41,x] = F[37,x]
F[42,x] = F[10,x]*F[30,x]
F[43,x] = F[10,x]*F[2,x]
F[44,x] = F[10,x]*F[45,x]
F[45,x] = F[13,x]+F[46,x]
F[46,x] = F[30,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[10,x]*F[14,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \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_{44}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{43}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = x^{2}
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{2}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{30}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_13(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_6(x))
Eq(F_13(x), F_14(x) + F_2(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_44(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_19(x) + F_6(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_10(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), 2*F_15(x) + F_24(x) + F_28(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_19(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_43(x))
Eq(F_31(x), F_10(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_10(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), x**2)
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), 2*F_15(x) + F_38(x) + F_42(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_34(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_10(x)*F_30(x))
Eq(F_43(x), F_10(x)*F_2(x))
Eq(F_44(x), F_10(x)*F_45(x))
Eq(F_45(x), F_13(x) + F_46(x))
Eq(F_46(x), F_30(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_10(x)*F_14(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, 2, 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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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