0123_0132_0231_0312_1203_1230
Counting sequence:
1, 1, 2, 6, 18, 54, 166, 527, 1724, 5781, 19770, 68677, 241616, 859003, 3081054, 11134870, 40505546, 148195362, 544949638, 2012979284, 7465880370, 27791334016, 103795058102, 388825802598, 1460606594398, 5500644720302, 20763951410102, 78549995674506, 297751596001566, 1130766611311282, 4301779598911286, 16391940054856327, 62556866313580356, 239079065308904917, 914941175383570418, 3505872956890561873, 13449920218742865824, 51657746023310607679, 198617422737362473190, 764436354256620221019, 2944999310166431879688, 11356077516621109055517, 43828002536257418140186, 169292384120445987235115, 654438786711269434015164, 2531809762603518271194253, 9801873981060437738088302, 37974290241796553457826641, 147217714653386287662790540, 571095135274636522739238191, 2216785519777151866286226954, 8609842749640896563925700105, 33459018899760955748493507368, 130097303955146830053791957799, 506117898018466559205877060918, 1969949415546263084956330712921, 7671328063757202309166702096140, 29887574946270884184507937215343, 116495290260046913279956366249922, 454272985836705023847542042852781, 1772188576332573813707610249384472, 6916420186476212140142500631788931, 27003835551688006076688233424111822, 105471881906750237450720595764700886, 412106872506176422379314148018623274, 1610793797622432916798250278982077362, 6298283500748796127424140412734039750, 24634972203012890669165731537332392588, 96388498029444861480363689509216244226, 377257128412085271322691264836870403464, 1477014710192496571993764711772039074902, 5784469004798880842217461585231931622890, 22660518056663018397512160993336792853078, 88797415543477945186994981285271772349066, 348057993390660237157455614967826630154694, 1364647447776169454364020862885423888465862, 5351848280286431601516523458219579614273990, 20994165287740728124881136359867475872865158, 82376258313834321780169950974777339906340886, 323304230745747667120785030468058403112037724, 1269182105997532402403845598616320046935235242, 4983531037693494875195449402320361495451644080, 19572604040924265761658976882937127126700205598, 76887528245678318651466757957403494995140814136, 302104191255500653823611179287532446271074759474, 1187268537374366978327283436152138017714739960652, 4666920453284000601456863131679035262982292227990, 18348434462590559224572269447713956652206139070668, 72152739535442228093075414984711102134233843013106, 283785311410099718338677016249783304064181029250024, 1116370688648636226206764261466006439108857631906542, 4392447414478469448681813459930178649375738478738460, 17285526697232892271308280977719078800833409273259978, 68035391337511961401976365294307640399781010079595896, 267831564070170983012891753000270313685097842814956614, 1054536596814785033641506585075878435579023236623638182, 4152723929046651064466611770867807469520555466317665150, 16355901804551040176886065461468155139430490768623866718, 64429465446209544968587205149555786939354261188644557686, 253841002455153834731766239710560625550704160862492584194, 1000241541104973637422851664770374101338849680121517647342
Generating function in Maple syntax:
1/2*((-2*x^2+2*x-1)*(1-4*x)^(1/2)-2*x^3+2*x^2-2*x+1)/x/(x-1)^2
Generating function in latex syntax:
\frac{\left(-2 x^{2}+2 x -1\right) \sqrt{1-4 x}-2 x^{3}+2 x^{2}-2 x +1}{2 x \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**3 + x**2 - x + sqrt(1 - 4*x)*(-2*x**2 + 2*x - 1)/2 + 1/2)/(x*(x - 1)**2)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^4*F(x)^2+(2*x^3-2*x^2+2*x-1)*(x-1)^2*F(x)+x^5+2*x^4-6*x^3+7*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x^{3}-2 x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{5}+2 x^{4}-6 x^{3}+7 x^{2}-4 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 54
a(n+4) = -4*(2*n+1)/(n+5)*a(n)+2*(9*n+10)/(n+5)*a(1+n)-8*(2*n+5)/(n+5)*a(n+2)+(26+7*n)/(n+5)*a(n+3)+2/(n+5), 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) = 18
a \! \left(5\right) = 54
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(1+n \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{2}{n +5}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22071/
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[4,x]*F[8,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[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[5,x]*F[8,x]
F[21,x] = F[22,x,1]
F[22,x,k[0]] = F[23,x,k[0]]+F[31,x,k[0]]
F[23,x,k[0]] = F[10,x]+F[24,x,k[0]]
F[24,x,k[0]] = F[25,x,k[0]]+F[29,x,k[0]]
F[25,x,k[0]] = F[26,x,k[0]]
F[26,x,k[0]] = F[27,x,k[0]]*F[28,x,k[0]]
F[27,x,k[0]] = k[0]*x
F[28,x,k[0]] = F[1,x]+F[25,x,k[0]]
F[29,x,k[0]] = F[30,x,k[0]]
F[30,x,k[0]] = F[24,x,k[0]]*F[8,x]
F[31,x,k[0]] = (F[32,x,k[0]]*k[0]-F[32,x,1])/(-1+k[0])
F[32,x,k[0]] = F[33,x,k[0]]
F[33,x,k[0]] = F[22,x,k[0]]*F[28,x,k[0]]*F[8,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_{4}\! \left(x \right) F_{8}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x , 1\right)
F_{22}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{31}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{10}\! \left(x \right)+F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{25}\! \left(x , y\right)+F_{29}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{26}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right) F_{28}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = y x
F_{28}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{24}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{31}\! \left(x , y\right) = \frac{y F_{32}\! \left(x , y\right)-F_{32}\! \left(x , 1\right)}{-1+y}
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{28}\! \left(x , y\right) F_{8}\! \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_4(x)*F_8(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_5(x)*F_8(x))
Eq(F_21(x), F_22(x, 1))
Eq(F_22(x, k_0), F_23(x, k_0) + F_31(x, k_0))
Eq(F_23(x, k_0), F_10(x) + F_24(x, k_0))
Eq(F_24(x, k_0), F_25(x, k_0) + F_29(x, k_0))
Eq(F_25(x, k_0), F_26(x, k_0))
Eq(F_26(x, k_0), F_27(x, k_0)*F_28(x, k_0))
Eq(F_27(x, k_0), k_0*x)
Eq(F_28(x, k_0), F_1(x) + F_25(x, k_0))
Eq(F_29(x, k_0), F_30(x, k_0))
Eq(F_30(x, k_0), F_24(x, k_0)*F_8(x))
Eq(F_31(x, k_0), (-k_0*F_32(x, k_0) + F_32(x, 1))/(1 - k_0))
Eq(F_32(x, k_0), F_33(x, k_0))
Eq(F_33(x, k_0), F_22(x, k_0)*F_28(x, k_0)*F_8(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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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"}}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[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": 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