0132_1203_1320_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 128, 296, 661, 1448, 3138, 6756, 14479, 30916, 65796, 139600, 295337, 623120, 1311366, 2753292, 5768099, 12059724, 25167112, 52430296, 109053629, 226494392, 469764298, 973081076, 2013268791, 4160752788, 8589938188, 17716744096, 36507226449, 75161932576, 154618828046, 317827585820, 652835035467, 1340029803420, 2748779077136, 5634997100712, 11544872100709, 23639500006984, 48378511632722, 98956046511236, 202310139523039, 413416372056932, 844424930146068, 1724034232367856, 3518437208899321, 7177611906138928, 14636698788972438, 29836347531349036, 60798594969522419, 123848989752710636, 252201579132771096, 513410357520261240, 1044835113549981197, 2125699024118901720, 4323455642275705306, 8791026472627238932, 17870283321406160519, 36317027395115713844, 73786976294838242332, 149879795598890144576, 304371277216207641249, 617965926469270020672, 1254378597012249553438, 2545650682171918168636, 5165088340638674500251, 10477750633867025367868, 21250649172913403513888, 43091594156185512629576, 87363779933088436510389, 177088743107611695573096, 358899852698093036302946, 727244438361925362973860, 1473378342655329306740527, 2984535617173615775126020, 6044629098073145873603876, 12240373923598120393975952, 24782979302099898081555529, 50170421514007110750388304, 101549768847628850675403942, 205517389334486959700138316, 415870481947432436099016259, 841412370451781905595593612, 1702167554017397877986394408, 3443020734262463889563291416, 6963412720980264046307679581, 14081567946871200626977647608, 28472620903563746322679970538, 57564211826770182782809393716, 116366363692825745840517798359, 235208607464222252230833727956, 475368975085586025561263831596, 960641470485455093321720531680, 1941089981599476271041826921457, 3921794044456084710880425684320, 7922816251426433759354395180846, 16004088827881396193895878119772, 32325090305819849738165931893739
Generating function in Maple syntax:
-(2*x^7-8*x^6+11*x^5-20*x^4+28*x^3-20*x^2+7*x-1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{2 x^{7}-8 x^{6}+11 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**7 + 8*x**6 - 11*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)+2*x^7-8*x^6+11*x^5-20*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+2 x^{7}-8 x^{6}+11 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/4*2^n*n+2^(-1+n)+1/6*n^3-n^2-1/6*n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{2^{n} n}{4}+2^{-1+n}+\frac{n^{3}}{6}-n^{2}-\frac{n}{6}+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) = 52
a(6) = 128
a(7) = 296
a(n+2) = -4*a(n)+4*a(n+1)+1/6*(n-3)*(n^2-9*n-4), n >= 8
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) = 52
a \! \left(6\right) = 128
a \! \left(7\right) = 296
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(n -3\right) \left(n^{2}-9 n -4\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19951/
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[27,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[26,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[10,x]
F[25,x] = F[20,x]
F[26,x] = F[12,x]*F[13,x]
F[27,x] = F[14,x]+F[28,x]
F[28,x] = F[21,x]+F[29,x]+F[39,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[38,x]
F[31,x] = F[32,x]+F[6,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[28,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[48,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[21,x]+F[43,x]+F[47,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[10,x]+F[36,x]
F[46,x] = F[42,x]
F[47,x] = F[12,x]*F[41,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[24,x]
F[52,x] = 2*F[21,x]+F[53,x]+F[55,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[36,x]
F[55,x] = F[12,x]*F[48,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_{27}\! \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_{20}\! \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_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{28}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{21}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{24}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{36}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \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_27(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_26(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_10(x))
Eq(F_25(x), F_20(x))
Eq(F_26(x), F_12(x)*F_13(x))
Eq(F_27(x), F_14(x) + F_28(x))
Eq(F_28(x), F_21(x) + F_29(x) + F_39(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_38(x))
Eq(F_31(x), F_32(x) + F_6(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_28(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_48(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_21(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_10(x) + F_36(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_12(x)*F_41(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_24(x))
Eq(F_52(x), 2*F_21(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_36(x))
Eq(F_55(x), F_12(x)*F_48(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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1, 3, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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