0132_0231_0321_1032_1302_2301
Counting sequence:
1, 1, 2, 6, 18, 50, 134, 354, 930, 2438, 6386, 16722, 43782, 114626, 300098, 785670, 2056914, 5385074, 14098310, 36909858, 96631266, 252983942, 662320562, 1733977746, 4539612678, 11884860290, 31114968194, 81460044294, 213265164690, 558335449778, 1461741184646, 3826888104162, 10018923127842, 26229881279366, 68670720710258, 179782280851410, 470676121843974, 1232246084680514, 3226062132197570, 8445940311912198, 22111758803539026, 57889336098704882, 151556249492575622, 396779412379021986, 1038781987644490338, 2719566550554449030, 7119917664018856754, 18640186441502121234, 48800641660487506950, 127761738539960399618, 334484573959393691906, 875691983338220676102, 2292591376055268336402, 6002082144827584333106, 15713655058427484662918, 41138883030454869655650, 107702994032937124304034, 281970099068356503256454, 738207303172132385465330, 1932651810448040653139538, 5059748128171989573953286, 13246592574067928068720322, 34680029594031794632207682, 90793496208027455827902726, 237700459030050572851500498, 622307880882124262726598770, 1629223183616322215328295814, 4265361669966842383258288674, 11166861826284204934446570210, 29235223808885772420081421958, 76538809600373112325797695666, 200381204992233564557311665042, 524604805376327581346137299462, 1373433211136749179481100233346, 3595694828033919957097163400578, 9413651272965010691810389968390, 24645258990861112118334006504594, 64522125699618325663191629545394, 168921118107993864871240882131590, 442241228624363268950531016849378, 1157802567765095941980352168416546, 3031166474670924556990525488400262, 7935696856247677728991224296784242, 20775924094072108629983147401952466, 54392075425968648160958217909073158, 142400302183833835852891506325267010, 372808831125532859397716301066727874, 976026191192764742340257396874916614, 2555269742452761367623055889558021970, 6689783036165519360528910271799149298, 17514079366043796713963674925839425926, 45852455061965870781362114505719128482, 120043285819853815630122668591317959522, 314277402397595576109005891268234750086, 822788921372932912696895005213386290738, 2154089361721203161981679124371924122130, 5639479163790676573248142367902386075654, 14764348129650826557762747979335234104834, 38653565225161803100040101570103316238850, 101196347545834582742357556730974714611718, 264935477412341945127032568622820827596306
Generating function in Maple syntax:
-(x^4+x^3+2*x^2-3*x+1)/(x-1)/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{x^{4}+x^{3}+2 x^{2}-3 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-x**4 - x**3 - 2*x**2 + 3*x - 1)/((x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)*F(x)+x^4+x^3+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{4}+x^{3}+2 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/5*(-8*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)-2+1/5*(8*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-8 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}-2+\frac{\left(8 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(n+2) = -a(n)+3*a(n+1)+2, n >= 5
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(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+2, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22321/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[10,x]*F[21,x]*F[7,x]
F[7,x] = F[11,x]+F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]*F[8,x]
F[10,x] = x
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[1,x]+F[13,x]
F[15,x] = F[16,x]+F[17,x]+F[20,x]
F[16,x] = 0
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[9,x]
F[20,x] = F[10,x]*F[11,x]
F[21,x] = F[22,x]+F[8,x]
F[22,x] = F[23,x]+F[25,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[1,x]+F[10,x]
F[25,x] = F[26,x]
F[26,x] = F[10,x]*F[14,x]*F[5,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_{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_{10}\! \left(x \right) F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{5}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_10(x)*F_21(x)*F_7(x))
Eq(F_7(x), F_11(x) + F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_8(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_1(x) + F_13(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_20(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_9(x))
Eq(F_20(x), F_10(x)*F_11(x))
Eq(F_21(x), F_22(x) + F_8(x))
Eq(F_22(x), F_23(x) + F_25(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_10(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_10(x)*F_14(x)*F_5(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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, 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