0132_0231_0312_1320_3021
Counting sequence:
1, 1, 2, 6, 19, 57, 172, 537, 1739, 5802, 19798, 68713, 241661, 859058, 3081120, 11134948, 40505637, 148195467, 544949758, 2012979420, 7465880523, 27791334187, 103795058292, 388825802808, 1460606594629, 5500644720555, 20763951410378, 78549995674806, 297751596001891, 1130766611311633, 4301779598911664, 16391940054856733, 62556866313580791, 239079065308905382, 914941175383570914, 3505872956890562401, 13449920218742866385, 51657746023310608274, 198617422737362473820, 764436354256620221685, 2944999310166431880391, 11356077516621109056258, 43828002536257418140966, 169292384120445987235935, 654438786711269434016025, 2531809762603518271195156, 9801873981060437738089248, 37974290241796553457827631, 147217714653386287662791575, 571095135274636522739239272, 2216785519777151866286228082, 8609842749640896563925701281, 33459018899760955748493508593, 130097303955146830053791959074, 506117898018466559205877062244, 1969949415546263084956330714299, 7671328063757202309166702097571, 29887574946270884184507937216828, 116495290260046913279956366251462, 454272985836705023847542042854377, 1772188576332573813707610249386125, 6916420186476212140142500631790642, 27003835551688006076688233424113592, 105471881906750237450720595764702716, 412106872506176422379314148018625165, 1610793797622432916798250278982079315, 6298283500748796127424140412734041766, 24634972203012890669165731537332394668, 96388498029444861480363689509216246371, 377257128412085271322691264836870405675, 1477014710192496571993764711772039077180, 5784469004798880842217461585231931625236, 22660518056663018397512160993336792855493, 88797415543477945186994981285271772351551, 348057993390660237157455614967826630157250, 1364647447776169454364020862885423888468490, 5351848280286431601516523458219579614276691, 20994165287740728124881136359867475872867933, 82376258313834321780169950974777339906343736, 323304230745747667120785030468058403112040650, 1269182105997532402403845598616320046935238245, 4983531037693494875195449402320361495451647161, 19572604040924265761658976882937127126700208758, 76887528245678318651466757957403494995140817376, 302104191255500653823611179287532446271074762795, 1187268537374366978327283436152138017714739964055, 4666920453284000601456863131679035262982292231476, 18348434462590559224572269447713956652206139074238, 72152739535442228093075414984711102134233843016761, 283785311410099718338677016249783304064181029253765, 1116370688648636226206764261466006439108857631910370, 4392447414478469448681813459930178649375738478742376, 17285526697232892271308280977719078800833409273263983, 68035391337511961401976365294307640399781010079599991, 267831564070170983012891753000270313685097842814960800, 1054536596814785033641506585075878435579023236623642460, 4152723929046651064466611770867807469520555466317669521, 16355901804551040176886065461468155139430490768623871183, 64429465446209544968587205149555786939354261188644562246, 253841002455153834731766239710560625550704160862492588850, 1000241541104973637422851664770374101338849680121517652095
Generating function in Maple syntax:
1/2*((-2*x^3+4*x^2-3*x+1)*(1-4*x)^(1/2)-2*x^5-2*x^4+4*x^3-4*x^2+3*x-1)/x/(x-1)^3
Generating function in latex syntax:
\frac{\left(-2 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}-2 x^{5}-2 x^{4}+4 x^{3}-4 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**5 - x**4 + 2*x**3 - 2*x**2 + 3*x/2 + sqrt(1 - 4*x)*(-2*x**3 + 4*x**2 - 3*x + 1)/2 - 1/2)/(x*(x - 1)**3)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2+(2*x^5+2*x^4-4*x^3+4*x^2-3*x+1)*(x-1)^3*F(x)+x^9+2*x^8-3*x^7+4*x^6-12*x^5+22*x^4-24*x^3+16*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(2 x^{5}+2 x^{4}-4 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+2 x^{8}-3 x^{7}+4 x^{6}-12 x^{5}+22 x^{4}-24 x^{3}+16 x^{2}-6 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 172
a(7) = 537
a(8) = 1739
a(n+4) = -4*(2*n+1)/(n+5)*a(n)+2*(9*n+10)/(n+5)*a(n+1)-8*(2*n+5)/(n+5)*a(n+2)+(26+7*n)/(n+5)*a(n+3)-1/2*(3*n^2-3*n+2)/(n+5), 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) = 57
a \! \left(6\right) = 172
a \! \left(7\right) = 537
a \! \left(8\right) = 1739
a \! \left(n +4\right) = -\frac{4 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{2 \left(9 n +10\right) a \! \left(n +1\right)}{n +5}-\frac{8 \left(2 n +5\right) a \! \left(n +2\right)}{n +5}+\frac{\left(26+7 n \right) a \! \left(n +3\right)}{n +5}-\frac{3 n^{2}-3 n +2}{2 \left(n +5\right)}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21666/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[16,x]+F[8,x]
F[8,x] = F[9,x]^2*F[13,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[9,x]^2*F[10,x]
F[18,x] = F[14,x]^2
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_{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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{16}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{13}\! \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_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14} \left(x \right)^{2}
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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_12(x)*F_7(x))
Eq(F_7(x), F_16(x) + F_8(x))
Eq(F_8(x), F_13(x)*F_9(x)**2)
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)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_10(x)*F_9(x)**2)
Eq(F_18(x), F_14(x)**2)
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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": [[{"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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": []}, "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 0], [0, 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": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], 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