0213_0231_2031_3120
Counting sequence:
1, 1, 2, 6, 20, 65, 200, 583, 1622, 4341, 11252, 28403, 70130, 169969, 405488, 954351, 2220014, 5111789, 11665388, 26410987, 59375594, 132644841, 294649832, 651165671, 1432354790, 3137339365, 6845104100, 14881390563, 32245809122, 69659000801, 150055419872, 322390982623, 690952863710, 1477468749789, 3152505995228, 6713033883611, 14267881357274, 30270929502169, 64115271794648, 135583527600087, 286285340082134, 603631883648981, 1271035441709012, 2672912767123411, 5614106371424210, 11777968556736465, 24681837020315600, 51668250412449743, 108051206684803022, 225742931321946061, 471189111013638092, 982629143696900043, 2047448980593311690, 4262657047306174409, 8867587666292506568, 18433233274827440071, 38289604031903956934, 79479526223834513349, 164867775158779117508, 341769168521892200387, 708037919016679899074, 1465939693107605929921, 3033336478620614393792, 6273045906565854396351, 12965755240808601092030, 26784672395026268946365, 55303338732981235744700, 114130005584040995848123, 235417347868681297723322, 485370730067445722120121, 1000256250652826726825912, 2060427526057062077300663, 4242455989048017518854070, 8731655626825974000123829, 17963882100836130392899508, 36943072995489234506743731, 75945097777509634326658994, 156064767525876035022225329, 320592015789054474267393968, 658335666643891499950931887, 1351427950601703588674666414, 2773275830195959326775967661, 5689204907106444896167264172, 11667343085100786165089304491, 23919806266895052850736398250, 49024359837012442292684849065, 100447228500140308867986751400, 205749503091852968501593505703, 421325155245533642935198810022, 862534722372088706535964802981, 1765302496020952272006151143332, 3611999549624918297086919704483, 7388645124474792170735422930850, 15110296119517351635418710278049, 30894031620405950140382544134048, 63149797284025818583154124914591, 129052773215422322897683902103454, 263671324847471715511314266718109, 538593048771968966960911774384028, 1099924580185531798811170662514587, 2245801494629336913426596816224154
Generating function in Maple syntax:
(x^5-12*x^4+23*x^3-19*x^2+7*x-1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
\frac{x^{5}-12 x^{4}+23 x^{3}-19 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(x**5 - 12*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**2*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)^3*F(x)-x^5+12*x^4-23*x^3+19*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)-x^{5}+12 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/16*(3*n^2-17*n+46)*2^n-n-2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(3 n^{2}-17 n +46\right) 2^{n}}{16}-n -2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+n-1, 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) = 20
a \! \left(5\right) = 65
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+n -1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21355/
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[23,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[21,x]+F[9,x]
F[9,x] = F[10,x]*F[15,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = F[10,x]+F[11,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]*F[15,x]*F[19,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[14,x]*F[19,x]
F[21,x] = F[22,x]
F[22,x] = F[11,x]*F[14,x]*F[17,x]*F[19,x]
F[23,x] = F[24,x]
F[24,x] = F[14,x]*F[25,x]
F[25,x] = F[26,x]+F[33,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[30,x]
F[30,x] = F[14,x]*F[19,x]*F[31,x]
F[31,x] = F[1,x]+F[32,x]
F[32,x] = F[29,x]
F[33,x] = F[34,x]
F[34,x] = F[13,x]*F[14,x]*F[35,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[1,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[14,x]*F[36,x]
F[40,x] = F[15,x]*F[29,x]
F[41,x] = F[15,x]*F[19,x]*F[29,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_{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_{14}\! \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_{15}\! \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_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \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_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{29}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{29}\! \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_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_14(x)*F_8(x))
Eq(F_8(x), F_21(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_15(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_14(x))
Eq(F_13(x), F_10(x) + F_11(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_18(x))
Eq(F_18(x), F_14(x)*F_15(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_14(x)*F_19(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_14(x)*F_17(x)*F_19(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_14(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_33(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_14(x)*F_19(x)*F_31(x))
Eq(F_31(x), F_1(x) + F_32(x))
Eq(F_32(x), F_29(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_13(x)*F_14(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_1(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_14(x)*F_36(x))
Eq(F_40(x), F_15(x)*F_29(x))
Eq(F_41(x), F_15(x)*F_19(x)*F_29(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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 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": [2, 0, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 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": [[2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 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": [2, 0, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 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", "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]]}]]}, 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