0213_0231_1230_1320_2301
Counting sequence:
1, 1, 2, 6, 19, 56, 153, 395, 978, 2345, 5482, 12558, 28301, 62942, 138487, 302029, 653908, 1407103, 3012160, 6419336, 13627463, 28830828, 60811237, 127918751, 268426374, 562025861, 1174392198, 2449458290, 5100255793, 10603179690, 22011683267, 45633999713, 94489248616, 195420975547, 403726884412, 833223608524, 1717986865483, 3539052992408, 7284264467345, 14980845853971, 30786325494778, 63221918504993, 129742371975522, 266081813809046, 545357767252629, 1117103813683766, 2286984185624143, 4679521487650485, 9570149207982908, 19562510881194935, 39969446692700152, 81627743245858768, 166633186212457231, 340021771866200452, 693554342614762237, 1414130282994018119, 2882303761516774958, 5872693914090758013, 11961560610295640830, 24355466784819216506, 49575624698093963193, 100880631653098621058, 205220027820018238107, 417357584667678045641, 848550227390638776784, 1724770570891842438771, 3504881374004814127988, 7120443212451886200980, 14462247353788287698323, 29367216565345605356080, 59619876846229269956137, 121010641123534657681723, 245563057109221550138146, 498209663942747569013657, 1010586427334104074639962, 2049507053565426021590814, 4155682504925287786834333, 8424701805439447059947918, 17076077202056637091368615, 34605501586468760124534941, 70117697537648492131452708, 142048783804718928026391023, 287724345068281743578402992, 582702245054251262206624536, 1179911599943878074511386839, 2388837419558507249217470876, 4835703278458516698822675733, 9787463435600037798419073775, 19807040628566084398383758038, 40078308771864186399856811125, 81085072573192408005890191222, 164027055205312886424131400578, 331767930528481913672962615361, 670963501292676108995322531162, 1356782283056776781289437225587, 2743275127056402689176456226609, 5545971375998503631548073335608, 11210784995768403769486465646123, 22659254479079600551753566326700, 45793877933244787129068399677276, 92538493816660746309259330223323
Generating function in Maple syntax:
(2*x^8-8*x^6+27*x^5-48*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{8}-8 x^{6}+27 x^{5}-48 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**8 - 8*x**6 + 27*x**5 - 48*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^5*F(x)+2*x^8-8*x^6+27*x^5-48*x^4+48*x^3-27*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+2 x^{8}-8 x^{6}+27 x^{5}-48 x^{4}+48 x^{3}-27 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-1/24*n^4+5/12*n^3-47/24*n^2+55/12*n+3/4*2^n*n-2^(n+1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -\frac{n^{4}}{24}+\frac{5 n^{3}}{12}-\frac{47 n^{2}}{24}+\frac{55 n}{12}+\frac{3 \,2^{n} n}{4}-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) = 56
a(6) = 153
a(7) = 395
a(8) = 978
a(n+2) = -4*a(n)+4*a(n+1)-1/24*(-1+n)*(n^3-17*n^2+90*n-192), n >= 9
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) = 56
a \! \left(6\right) = 153
a \! \left(7\right) = 395
a \! \left(8\right) = 978
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{\left(-1+n \right) \left(n^{3}-17 n^{2}+90 n -192\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21890/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[15,x]*F[7,x]
F[7,x] = F[24,x]+F[8,x]
F[8,x] = F[21,x]*F[9,x]
F[9,x] = F[10,x]+F[16,x]
F[10,x] = F[11,x]+F[16,x]
F[11,x] = F[12,x]^2
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = x
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[13,x]*F[15,x]*F[18,x]
F[18,x] = F[12,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[15,x]*F[18,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[15,x]*F[21,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[15,x]*F[26,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[15,x]*F[9,x]
F[31,x] = F[22,x]*F[32,x]
F[32,x] = F[13,x]+F[28,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_{15}\! \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_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{24}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{21}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12} \left(x \right)^{2}
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \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_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \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_15(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_15(x)*F_7(x))
Eq(F_7(x), F_24(x) + F_8(x))
Eq(F_8(x), F_21(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_16(x))
Eq(F_10(x), F_11(x) + F_16(x))
Eq(F_11(x), F_12(x)**2)
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_13(x)*F_15(x)*F_18(x))
Eq(F_18(x), F_12(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_15(x)*F_18(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_15(x)*F_21(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_15(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_15(x)*F_9(x))
Eq(F_31(x), F_22(x)*F_32(x))
Eq(F_32(x), F_13(x) + F_28(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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", 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