0132_0231_0321_2013

Counting sequence:
1, 1, 2, 6, 20, 65, 208, 663, 2113, 6736, 21476, 68472, 218309, 696032, 2219149, 7075281, 22558018, 71921410, 229306016, 731093133, 2330933912, 7431683675, 23694332113, 75544304472, 240856839136, 767920458916, 2448349954841, 7806039586224, 24887885778425, 79349694768785, 252989511280930, 806602886182014, 2571680591432476, 8199252913233577, 26141562276099824, 83346773842313055, 265733342045618801, 847233861844869440, 2701223757361034236, 8612273559797460480, 27458390170997227453, 87545197623802432048, 279119117299376540549, 889911540056893553985, 2837292396124244253794, 9046099279249475325818, 28841550585981438636424, 91955108442358466376021, 293179173686863269174424, 934739019285608591531875, 2980215215110184463616673, 9501778084713133815421512, 30294384886494453731542328, 96587159526234206535053324, 307947476745276783224624497, 981824591373612100840301856, 3130337479671346713953929585, 9980410780836061543054490593, 31820402752449094914421099970, 101452540738333153843315159542, 323459702956492738887005758436, 1031282003144256650475553145681, 3288021847198330633998931566928, 10483153623055378477299387748647, 33423290656727271396563315197921, 106563005608085537438829582258352, 339753325932414044676926702755220, 1083230731185182216265080767192392, 3453649243208299539864935857960181, 11011221110818151092780257415277120, 35106920770765302012468229557175549, 111930899725000236527662162608023409, 356867706941731490705889155033804290, 1137796270473508706345818619283599602, 3627619781564605891805737404198184496, 11565888921504629175419045076583766877, 36875360318739939223310675868057917912, 117569190562483983965137124265962155115, 374844189996785353257841251940456818097, 1195110437539934029534683355908582491832, 3810353731050603981553657467946979586064, 12148496992141883174833558180268589841972, 38732881402952440137465784925082459919625, 123491498803974649036450805983379632230480, 393726201730233762889793084806486718591081, 1255311688904105427762119264381159027434481, 4002292530632647511553014394013638046114786, 12760452756352482174851943673700941452977262, 40683971324146325355400643861950497419590252, 129712131247064607836361878085093962392978489, 413559356302804584965650538822238936812235248, 1318545455550523323883769108569325298942573039, 4203899855864890473829887339757628845146155153, 13403234544357861116470254450229872382962115232, 42733343421690100953240775429038167232164803756, 136246115365102234261265994813790005923861597520, 434391565595561385775641794758033023227060249197, 1384964494252986802388480224470030114751783123216, 4415662739012055387441502283848849562481978146549, 14078395154249924762637476251557130934823485063265, 44885948459812081076389520173910038189826849185826

Generating function in Maple syntax:
-(x^2-3*x+1)/(x^4+x^3-3*x^2+4*x-1)

Generating function in latex syntax:
-\frac{x^{2}-3 x +1}{x^{4}+x^{3}-3 x^{2}+4 x -1}

Generating function in sympy syntax:
(-x**2 + 3*x - 1)/(x**4 + x**3 - 3*x**2 + 4*x - 1)

Implicit equation for the generating function in Maple syntax:
(x^4+x^3-3*x^2+4*x-1)*F(x)+x^2-3*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{4}+x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)+x^{2}-3 x +1 = 0

Explicit closed form in Maple syntax:
-35/2687*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4+_Z^3-3*_Z^2+4*_Z-1))+76/2687*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4+_Z^3-3*_Z^2+4*_Z-1))+862/2687*Sum(_alpha^(-n),_alpha = RootOf(_Z^4+_Z^3-3*_Z^2+4*_Z-1))-110/2687*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4+_Z^3-3*_Z^2+4*_Z-1))

Explicit closed form in latex syntax:
-\frac{35 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2687}+\frac{76 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{2687}+\frac{862 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2687}-\frac{110 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2687}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)+a(n+1)-3*a(n+2)+4*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) = a \! \left(n \right)+a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15313/
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[17,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[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[51,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[27,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]
F[27,x] = F[18,x]+F[28,x]
F[28,x] = 2*F[19,x]+F[29,x]+F[33,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[23,x]
F[32,x] = F[28,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[19,x]+F[37,x]+F[49,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[11,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[26,x]*F[4,x]
F[42,x] = F[36,x]+F[43,x]
F[43,x] = 2*F[19,x]+F[44,x]+F[48,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[40,x]
F[47,x] = F[43,x]
F[48,x] = F[35,x]*F[4,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[2,x]+F[36,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[50,x]+F[53,x]
F[53,x] = F[36,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[36,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_{17}\! \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_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \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_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{40}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{36}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{36}\! \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_17(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_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_51(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_27(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x))
Eq(F_27(x), F_18(x) + F_28(x))
Eq(F_28(x), 2*F_19(x) + F_29(x) + F_33(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_23(x))
Eq(F_32(x), F_28(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_49(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_11(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_26(x)*F_4(x))
Eq(F_42(x), F_36(x) + F_43(x))
Eq(F_43(x), 2*F_19(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_40(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_35(x)*F_4(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_2(x) + F_36(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_50(x) + F_53(x))
Eq(F_53(x), F_36(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_36(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": [2, 0, 1, 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": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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