0132_0231_1230_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 60, 187, 576, 1759, 5340, 16147, 48696, 146599, 440820, 1324507, 3977616, 11941039, 35839500, 107551267, 322719336, 968289079, 2905129380, 8715912427, 26148785856, 78448454719, 235349558460, 706057063987, 2118187969176, 6354597461959, 19063859494740, 57191712701947, 171575406541296, 514726756494799, 1544181343226220, 4632546177162307, 13897642826454216, 41692937069297239, 125078828387760900, 375236519523021067, 1125709627288539936, 3377129019304573279, 10131387332791626780, 30394162548130694227, 91182488743903710456, 273547468430734386919, 820642409690249671860, 2461927237866842037787, 7385781731192712157776, 22157345228762508562159, 66472035756656269864140, 199416107410706297947747, 598248322513593870553896, 1794744968103731565082999, 5384234905437094602091620, 16152704718563083619960107, 48458114160192850487250816, 145374342489585750716493439, 436123027486771650658962300, 1308369082496343748995850867, 3925107247561088841025480536, 11775321742827381711152297479, 35325965228770375509608604180, 105977895686887587281129236027, 317933687061815683347994555056, 953801061187752893053197359119, 2861403183567870365178019465260, 8584209550712834467570913171587, 25752628652156950146786449066376, 77257885956507743928506766302359, 231773657869597018761815137113540, 695320973608938630238035087753547, 2085962920827111038619284616086496, 6257888762481923411668212553911199, 18773666287446950826625355073037020, 56320998862343213663117500041717907, 168962996587034363355835369770367416, 506888989761112534800471848601529639, 1520666969283356493867347024385443700, 4562000907850107260533904030318040667, 13686002723550397339465438005277541136, 41058008170651343134123765844479461679, 123174024511954331633826201190732061580, 369522073535863599364388410886783537827, 1108566220607592007018984847289525319656, 3325698661822778438908593771126925371319, 9977095985468340152429059771897474938660, 29931287956405030128693736232725822465387, 89793863869215109728894322532244262694976, 269381591607645367872309195264866378682559, 808144774822936180988180041130866317242940, 2424434324468808697707045034065133314119347, 7273302973406426402606144923540468667139096, 21819908920219279826788454413311543450979399, 65459726760657840718305402525314905252062420, 196379180281973524630796286146705265554435707, 589137540845920578844149015581636896259804016, 1767412622537761746435967361027952887972405839, 5302237867613285259114942711649943062303205100, 15906713602839855816958909392081997983681590467, 47720140808519567530104890690510331544588721736, 143160422425558702748770997100059669820854065879
Generating function in Maple syntax:
(x^4+x^3-7*x^2+5*x-1)/(x-1)/(3*x-1)/(2*x-1)
Generating function in latex syntax:
\frac{x^{4}+x^{3}-7 x^{2}+5 x -1}{\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right)}
Generating function in sympy syntax:
(x**4 + x**3 - 7*x**2 + 5*x - 1)/((x - 1)*(2*x - 1)*(3*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(3*x-1)*(2*x-1)*F(x)+x^4+x^3-7*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) F \! \left(x \right)+x^{4}+x^{3}-7 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/2-1/4*2^n+5/18*3^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{1}{2}-\frac{2^{n}}{4}+\frac{5 \,3^{n}}{18} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+2) = -6*a(n)+5*a(n+1)+1, 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 +2\right) = -6 a \! \left(n \right)+5 a \! \left(n +1\right)+1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19047/
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[12,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[18,x]+F[30,x]+F[66,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[44,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[18,x]+F[19,x]+F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[13,x]+F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[14,x]+F[37,x]
F[40,x] = 2*F[18,x]+F[41,x]+F[43,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[17,x]+F[40,x]
F[43,x] = F[12,x]*F[36,x]
F[44,x] = F[29,x]+F[45,x]
F[45,x] = 2*F[18,x]+F[46,x]+F[56,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[53,x]
F[48,x] = F[18,x]+F[49,x]+F[51,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[20,x]+F[47,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[23,x]+F[48,x]
F[53,x] = 2*F[18,x]+F[46,x]+F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[27,x]+F[53,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[55,x]+F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[27,x]+F[59,x]
F[62,x] = 3*F[18,x]+F[63,x]+F[65,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[53,x]+F[62,x]
F[65,x] = F[12,x]*F[58,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[52,x]+F[55,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{66}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{44}\! \left(x \right) = F_{29}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{23}\! \left(x \right)+F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{27}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{27}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{53}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_22(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_18(x) + F_30(x) + F_66(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_44(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_18(x) + F_19(x) + F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_13(x) + F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_14(x) + F_37(x))
Eq(F_40(x), 2*F_18(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_17(x) + F_40(x))
Eq(F_43(x), F_12(x)*F_36(x))
Eq(F_44(x), F_29(x) + F_45(x))
Eq(F_45(x), 2*F_18(x) + F_46(x) + F_56(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_53(x))
Eq(F_48(x), F_18(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_20(x) + F_47(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_23(x) + F_48(x))
Eq(F_53(x), 2*F_18(x) + F_46(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_27(x) + F_53(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_55(x) + F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_27(x) + F_59(x))
Eq(F_62(x), 3*F_18(x) + F_63(x) + F_65(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_53(x) + F_62(x))
Eq(F_65(x), F_12(x)*F_58(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_52(x) + F_55(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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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