0213_0231_1230_1302_2031

Counting sequence:
1, 1, 2, 6, 19, 58, 172, 500, 1431, 4043, 11300, 31298, 86027, 234930, 638035, 1724642, 4642919, 12455585, 33313753, 88867705, 236523475, 628263682, 1665929798, 4410755654, 11662525642, 30801113337, 81263592126, 214207697812, 564196536478, 1484988113949, 3906147154001, 10269249578896, 26984904889140, 70879296112559, 186103934574034, 488481539115329, 1281780779804176, 3362536126991441, 8819026713661509, 23125239385045503, 60628071836212953, 158924960037932111, 416532763321652892, 1091570745064249016, 2860266065986868184, 7494078909769116118, 19633250360356064087, 51431897141799336310, 134723412437390829196, 352880092839547890128, 924246423906765593615, 2420625353575658643483, 6339410868291833006792, 16601748294887889988990, 43475461140362626919975, 113847016205417837517402, 298117618635050802406759, 780626800394569230178570, 2044043987897658510123943, 5352158898122101016742365, 14013952478950958615718165, 36693231627908126265117917, 96073955934222135086999963, 251547730501347223000034414, 658613624868096663697249382, 1724396337450505736646635022, 4514815284409065640950126766, 11820607211462620672214155997, 30948302841144543631290940418, 81027315300245410147995888284, 212140649756216065640231157466, 555410922609786358805644139581, 1454129984666106345441647491105, 3807067060728853868865454758244, 9967275841292023643075592978824, 26095236202964538968455291796339, 68319538730031335034794000602066, 178865951040916799260605628151301, 468284291373761266967015814724224, 1226000817887793422789305780974913, 3209750463822490655463665060312593, 8403325665582261799575202158973971, 22000401100749285487931730917791157, 57598283457819387767796925758873151, 150795392692825090347727442902573224, 394790087807126388319513826339668928, 1033579969270095883601854077540590752, 2705961672677379743542409338666087370, 7084332602892921380534963592850956211, 18547100191596794752026485613120474338, 48557116883115499164587521456604259180, 127124596634382554560674412828490176668, 332817477783167429002584003264487194807, 871329707562632601058599488272694615675, 2281175994097940787866243242146047082540, 5972208385379549500900963773700421993162, 15635472666448116911183898395957866452099, 40934264255085995638742414835253417984930, 107167447124011946329141870795672386115883, 280568372414730996225456938126415095507650, 734538356604261349776073365441265595224855

Generating function in Maple syntax:
(x^6-4*x^5+10*x^4-20*x^3+18*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)

Generating function in latex syntax:
\frac{x^{6}-4 x^{5}+10 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}

Generating function in sympy syntax:
(x**6 - 4*x**5 + 10*x**4 - 20*x**3 + 18*x**2 - 7*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)-x^6+4*x^5-10*x^4+20*x^3-18*x^2+7*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-x^{6}+4 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/2760*(92*((I-3/46*23^(1/2))*3^(1/2)-9/46*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-920-23*2^(2/3)*((I-21/23*23^(1/2))*3^(1/2)+63/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(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)+1/2760*(23*2^(2/3)*((I+21/23*23^(1/2))*3^(1/2)+63/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)-920-92*((I+3/46*23^(1/2))*3^(1/2)-9/46*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-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)+1/2760*((-42*3^(1/2)*2^(2/3)*23^(1/2)-46*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-920+(12*23^(1/2)*3^(1/2)*2^(1/3)-184*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)+1/2760*(828*5^(1/2)+1380)*(3/2-1/2*5^(1/2))^(-n)+1/2760*(-828*5^(1/2)+1380)*(3/2+1/2*5^(1/2))^(-n)+2^(n-1))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(92 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{46}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{46}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-920-23 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{21 \sqrt{23}}{23}\right) \sqrt{3}+\frac{63 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{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}}{2760}\\+\\\frac{\left(23 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{21 \sqrt{23}}{23}\right) \sqrt{3}+\frac{63 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-920-92 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{46}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{46}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{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}}{2760}\\+\\\frac{\left(\left(-42 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-46 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-920+\left(12 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}-184 \,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}}{2760}\\+\frac{\left(828 \sqrt{5}+1380\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{2760}+\frac{\left(-828 \sqrt{5}+1380\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{2760}+\\2^{n -1} & \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(5) = 58
a(6) = 172
a(n+6) = -2*a(n)+11*a(n+1)-25*a(n+2)+34*a(n+3)-24*a(n+4)+8*a(n+5), n >= 7

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(5\right) = 58
a \! \left(6\right) = 172
a \! \left(n +6\right) = -2 a \! \left(n \right)+11 a \! \left(n +1\right)-25 a \! \left(n +2\right)+34 a \! \left(n +3\right)-24 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21884/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[21,x]+F[8,x]
F[8,x] = F[15,x]*F[18,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[14,x]*F[9,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,x]*F[14,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]*F[9,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]*F[18,x]
F[21,x] = F[12,x]^2*F[2,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{21}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \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_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{2}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_21(x) + F_8(x))
Eq(F_8(x), F_15(x)*F_18(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_14(x)*F_9(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x)*F_9(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x)*F_18(x))
Eq(F_21(x), F_12(x)**2*F_2(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, 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, 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": [[{"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, 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, 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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 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