0213_0231_1203_1230_1302_2031

Counting sequence:
1, 1, 2, 6, 18, 51, 139, 369, 960, 2457, 6204, 15490, 38311, 93997, 229051, 554878, 1337393, 3209298, 7671826, 18278339, 43421559, 102887153, 243242408, 573928077, 1351818056, 3179151974, 7466468143, 17514553353, 41041533987, 96082057094, 224751398485, 525346207766, 1227181286174, 2864998082435, 6685305235555, 15592829275457, 36354479816320, 84730589858145, 197419236600932, 459855384539370, 1070901379285847, 2493367542879381, 5804202568558587, 13509143505015326, 31437681688247033, 73150637669774506, 170191246254111274, 395925649390075939, 920984897982890175, 2142191842910895137, 4982360969505600264, 11587413664119316885, 26947310720628832976, 62664946926872551502, 145719158200870898911, 338839067764186456273, 787874438206883866883, 1831930049671164446278, 4259420772319196520861, 9903375534131222937934, 23025448892841615984374, 53533556011005988058083, 124462369153502809919243, 289364179305787930889297, 672739402031016186837696, 1564029462375784167401129, 3636130559695734590606028, 8453390362701923113618898, 19652585067132664161352327, 45688414754519114225589949, 106216120823140551643116795, 246929504476785613828594686, 574055607011086488479697089, 1334546068995717240886957890, 3102501359923351621159604290, 7212558454344502110854957827, 16767402884475352247295956935, 38979966171612550843309441425, 90618406778868709289425559592, 210664220924596828194002990365, 489738913260078379333727477912, 1138512976012599441791445800886, 2646738016163344348507219852047, 6152952699398579352142457938521, 14303947025691892230258892277923, 33252778870988168807737180246086, 77303608031923553489570962407909, 179709643757077997891483614408134, 417775364504965178036051887101390, 971212174228656450714837767088067, 2257798996510729939432183476451699, 5248755201106320441175774633439905, 12201894330321619498596024779214016, 28366000986338880580650237401414257, 65943028920601608884187632437217396, 153299099199632495671745112503087418, 356377483380280849762044190264589623, 828477770886434723837418838278004133, 1925978435450406395964133166107299771, 4477359248469734075154130249015226654, 10408602686031112384852760523850551689

Generating function in Maple syntax:
(x^5-3*x^4+11*x^3-13*x^2+6*x-1)/(2*x-1)^2/(x^3-2*x^2+3*x-1)

Generating function in latex syntax:
\frac{x^{5}-3 x^{4}+11 x^{3}-13 x^{2}+6 x -1}{\left(2 x -1\right)^{2} \left(x^{3}-2 x^{2}+3 x -1\right)}

Generating function in sympy syntax:
(x**5 - 3*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((2*x - 1)**2*(x**3 - 2*x**2 + 3*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x^3-2*x^2+3*x-1)*F(x)-x^5+3*x^4-11*x^3+13*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4600*(-161*2^(1/3)*((I-3/161*23^(1/2))*3^(1/2)-9/161*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+4600-115*((I+9/23*23^(1/2))*3^(1/2)-27/23*I*23^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(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)+1/4600*(115*2^(2/3)*((I-9/23*23^(1/2))*3^(1/2)-27/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+161*((I+3/161*23^(1/2))*3^(1/2)-9/161*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-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)+1/4600*((90*2^(2/3)*23^(1/2)*3^(1/2)-230*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+(-6*23^(1/2)*3^(1/2)*2^(1/3)+322*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)-1/4*2^n*(n+9))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-161 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{161}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{161}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+4600-115 \left(\left(\mathrm{I}+\frac{9 \sqrt{23}}{23}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{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}}{4600}\\+\\\frac{\left(115 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{23}}{23}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+161 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{161}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{161}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{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}}{4600}\\+\\\frac{\left(\left(90 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-230 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+\left(-6 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+322 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{4600}\\-\frac{2^{n} \left(n +9\right)}{4} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 51
a(n+5) = 4*a(n)-12*a(n+1)+21*a(n+2)-18*a(n+3)+7*a(n+4), n >= 6

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) = 18
a \! \left(5\right) = 51
a \! \left(n +5\right) = 4 a \! \left(n \right)-12 a \! \left(n +1\right)+21 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22568/
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[8,x]
F[4,x] = F[23,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = 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[8,x]
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[11,x]
F[19,x] = F[14,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[20,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[28,x]*F[5,x]*F[8,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[25,x]*F[8,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[8,x]
F[32,x] = F[1,x]+F[30,x]
F[33,x] = F[15,x]+F[34,x]+F[36,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[11,x]+F[33,x]
F[36,x] = F[29,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{23}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{28}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right) F_{8}\! \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_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_23(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_8(x)*F_9(x))
Eq(F_8(x), 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_8(x))
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_11(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_20(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_28(x)*F_5(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_25(x)*F_8(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_8(x))
Eq(F_32(x), F_1(x) + F_30(x))
Eq(F_33(x), F_15(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_11(x) + F_33(x))
Eq(F_36(x), F_29(x)*F_8(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "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, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 3, 1], "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, 3], "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, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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