021_1023_1203
Counting sequence:
1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217, 20330163, 47261895, 109870576, 255418101, 593775046, 1380359512, 3208946545, 7459895657, 17342153393, 40315615410, 93722435101, 217878227876, 506505428836, 1177482265857, 2737314167775, 6363483400447, 14793304131648, 34390259761825, 79947654422626, 185855747875876, 432062194544201, 1004422742303477, 2334999585697905, 5428215467030962, 12619069972000553, 29335778567637640, 68197411225942776, 158539746514553601, 368560195659412891, 856798505175074247, 1991814870720950560, 4630407797472116077, 10764392156149521358, 25024175744225282480, 58174150717848920801, 135238492821245718801, 314391352772264597281, 730871223392151275042, 1699069457453170349365, 3949857278347473095292, 9182304143528229862188, 21346267331342913745345, 49624050985319754606951, 115361922436801666192351, 268183932671108403108496, 623452004125041631547737, 1449350069469709754618570, 3369330132830154403868732, 7832742263676085333916793, 18208916594837656948631485, 42330595389990954581929601, 98406695243973635182442626, 228767811546776653332100161, 531820639542373644213344832, 1236332990777541261158276800, 2874125504794653148380240897, 6681531172371250567037513923, 15532675498301986665510336775, 36109089654958112010836223376, 83943449140641613268525510501, 195144843610310602449414421526, 453656722204606692822028466952, 1054623928533840486835782068305, 2451703184802618677312703692537, 5699518419544781751088575407953, 13249772817562948385476100907090, 30801984798401900331563855597901, 71605927179624585974827940387476, 166463584759632905646832210873716, 386980884718051445322404607444097, 899621411814513110648377340972335, 2091366050767069346947155018902527, 4861836213390233264867114982207008, 11302397950451074211355412249788305, 26274887475339825451279161803853426, 61081702738507561195993775894190676, 141997731215294106896778416324653481, 330104675644207023749626858989432517, 767400267240540418651317520213181265, 1783989181648501315351477258985332242
Generating function in Maple syntax:
-(x-1)^2/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right)^{2}}{x^{3}-2 x^{2}+3 x -1}
Generating function in sympy syntax:
-(x - 1)**2/(x**3 - 2*x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
1/300*(((I+4/23*23^(1/2))*3^(1/2)-12/23*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+100-5/2*((I+1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3))*((-3/200*((I+5/207*23^(1/2))*3^(1/2)-133/207*I*23^(1/2)+5/9)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/20*2^(2/3)*((I-4/69*23^(1/2))*3^(1/2)+4/69*I*23^(1/2)+1/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)-1/138*I*23^(1/2)+5/6)*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+(-1/300*2^(1/3)*((I+32/23*23^(1/2))*3^(1/2)-37/23*I*23^(1/2)-8)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/30*2^(2/3)*((I+2/23*23^(1/2))*3^(1/2)-2/23*I*23^(1/2)+2)*(11+3*23^(1/2)*3^(1/2))^(1/3)+1/138*I*23^(1/2)+5/6)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n))
Explicit closed form in latex syntax:
\frac{\left(\left(\left(i+\frac{4 \sqrt{23}}{23}\right) \sqrt{3}-\frac{12 i \sqrt{23}}{23}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+100-\frac{5 \left(\left(i+\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 i \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}\right) \left(\left(-\frac{3 \left(\left(i+\frac{5 \sqrt{23}}{207}\right) \sqrt{3}-\frac{133 i \sqrt{23}}{207}+\frac{5}{9}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{200}+\frac{2^{\frac{2}{3}} \left(\left(i-\frac{4 \sqrt{23}}{69}\right) \sqrt{3}+\frac{4 i \sqrt{23}}{69}+\frac{1}{3}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{20}-\frac{i \sqrt{23}}{138}+\frac{5}{6}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(i-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 i \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{i \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}+\left(-\frac{2^{\frac{1}{3}} \left(\left(i+\frac{32 \sqrt{23}}{23}\right) \sqrt{3}-\frac{37 i \sqrt{23}}{23}-8\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}+\frac{2^{\frac{2}{3}} \left(\left(i+\frac{2 \sqrt{23}}{23}\right) \sqrt{3}-\frac{2 i \sqrt{23}}{23}+2\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{30}+\frac{i \sqrt{23}}{138}+\frac{5}{6}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}+\left(-\frac{11 \left(\left(i+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 i \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{i \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}\right)}{300}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+3) = a(n)-2*a(n+1)+3*a(n+2), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right), \quad n \geq 3
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/5310/
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[6,x]+F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[6,x]
F[9,x] = F[10,x]+F[2,x]
F[10,x] = F[11,x]+F[12,x]+F[25,x]
F[11,x] = 0
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]+F[4,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[18,x]+F[20,x]
F[18,x] = F[11,x]+F[12,x]+F[19,x]
F[19,x] = F[2,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[19,x]
F[24,x] = F[21,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[25,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_{6}\! \left(x \right)+F_{9}\! \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_{6}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{2}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{25}\! \left(x \right)
F_{11}\! \left(x \right) = 0
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
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_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \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_6(x) + F_9(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_6(x))
Eq(F_9(x), F_10(x) + F_2(x))
Eq(F_10(x), F_11(x) + F_12(x) + F_25(x))
Eq(F_11(x), 0)
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x) + F_4(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_20(x))
Eq(F_18(x), F_11(x) + F_12(x) + F_19(x))
Eq(F_19(x), F_2(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_21(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_25(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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 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": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[1, 2], [1, 2]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], 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