0213_0231_1302_3120
Counting sequence:
1, 1, 2, 6, 20, 64, 193, 552, 1511, 3990, 10229, 25588, 62707, 151026, 358385, 839664, 1945583, 4464622, 10158061, 22937580, 51445739, 114687978, 254279657, 560988136, 1232076775, 2694840294, 5872025573, 12750684132, 27598520291, 59559116770, 128177930209, 275146342368, 589215825887, 1258962288606, 2684354559965, 5712306503644, 12133282611163, 25726854103002, 54460185313241, 115105123532760, 242923350261719, 511960101683158, 1077521395220437, 2264993953218516, 4755387790131155, 9972570463928274, 20890720927743953, 43716582320373712, 91391406500741071, 190875218581913550, 398287092045578189, 830351181296435148, 1729663731886981067, 3600064952129290186, 7487234380503449545, 15559936712565063624, 32313327326383308743, 67058598451546685382, 139071156493200916421, 288230376151711743940, 596997166604232949699, 1235787737750464102338, 2556603436465683169217, 5286145098622393384896, 10923931256149875097535, 22562673845155995320254, 46578028786116617830333, 96107536624026763919292, 198210265072009132113851, 408595381232666568294330, 841909399524103935754169, 1733993942928697851903928, 3569813913144272428728247, 7346231359914091835555766, 15111572745182864683827125, 31073171457282265506119604, 63870006680811951515238323, 131234564558947440488611762, 269552678842199348797767601, 553461351792322419045171120, 1136012481119121852606709679, 2330960095944456877480345518, 4781301616575858385962925997, 9804388397074642606867087276, 20098391751093210029490175915, 41188102674270415982239416234, 84383022209101116394491084713, 172828035172107386815994920872, 353876765917594252019991838631, 724388351113085801483940659110, 1482433197044242879191701258149, 3032953096249181673502854348708, 6203627021868861902588236267427, 12685790552576934367028775485346, 25934843823028716759136652492705, 53008592482199983371180499271584, 108319753437470774053673369599903, 221294161422654577940967446282142, 451996667143878045971168236666781, 923008093291179532964787021414300, 1884441845401777269662442858741659
Generating function in Maple syntax:
-(x^6-2*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^{6}-2 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**6 + 2*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^6-2*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^{6}-2 x^{5}+12 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/32*(5*n^2-25*n+70)*2^n-n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(5 n^{2}-25 n +70\right) 2^{n}}{32}-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) = 20
a(5) = 64
a(6) = 193
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+n-2, 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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 193
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+n -2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21346/
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[10,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[11,x]*F[16,x]
F[10,x] = x
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[10,x]*F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[11,x]*F[12,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]*F[12,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[22,x]+F[36,x]
F[22,x] = F[23,x]+F[4,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[27,x]
F[25,x] = F[26,x]
F[26,x] = F[10,x]*F[11,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[10,x]*F[27,x]
F[31,x] = F[32,x]*F[33,x]
F[32,x] = F[17,x]
F[33,x] = F[1,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]*F[12,x]*F[33,x]
F[36,x] = F[37,x]
F[37,x] = F[34,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]*F[41,x]
F[40,x] = F[1,x]+F[25,x]
F[41,x] = F[16,x]+F[32,x]
F[42,x] = F[43,x]
F[43,x] = F[10,x]*F[12,x]*F[25,x]*F[32,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{16}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \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_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \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_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{34}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{25}\! \left(x \right) F_{32}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_19(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_11(x)*F_16(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_10(x)*F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_11(x)*F_12(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)*F_12(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_36(x))
Eq(F_22(x), F_23(x) + F_4(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_27(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_10(x)*F_11(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_10(x)*F_27(x))
Eq(F_31(x), F_32(x)*F_33(x))
Eq(F_32(x), F_17(x))
Eq(F_33(x), F_1(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x)*F_12(x)*F_33(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_34(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x)*F_41(x))
Eq(F_40(x), F_1(x) + F_25(x))
Eq(F_41(x), F_16(x) + F_32(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_10(x)*F_12(x)*F_25(x)*F_32(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, 3, 0, 2], "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": [1, 3, 0, 2], "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": [1, 3, 0, 2], "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, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 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": [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]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 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]]}, 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