0132_0231_1032_3021
Counting sequence:
1, 1, 2, 6, 20, 68, 235, 824, 2924, 10479, 37868, 137818, 504663, 1857895, 6872089, 25525565, 95167582, 356014169, 1335891631, 5026679699, 18962402752, 71699632133, 271689027610, 1031545376546, 3923764679359, 14950646900849, 57057041571827, 218074845309516, 834657180643571, 3198747192260146, 12274032023948820, 47151923399272702, 181337776554598396, 698116781763887168, 2690265004069397719, 10376893626818534899, 40061275557784696839, 154792021038790104976, 598578882290622843707, 2316472188513973276596, 8971201978961478632066, 34767863012312210635085, 134832961850700413079439, 523229955685563354857651, 2031683615323731179213362, 7893610983708708652991147, 30686077880222058983332402, 119355799126330409230136828, 464486560558140266425038643, 1808513614920135245166814529, 7045006562486652040630020565, 27456453196032069990627019922, 107054199795030483768168147469, 417592251898065276433109849354, 1629611789758877946158284135222, 6361986888274585491045142993499, 24846913854525876575132892661069, 97077298769564300660049207227009, 379422599212390807291910679767002, 1483487789042314103740000287938093, 5802234103493917377387991978367302, 22701389979238702254210546138666638, 88848730718307411995636548057985359, 347846327711299255452349783221733544, 1362250750813341057620612257065246652, 5336497632588629506529367146777493143, 20911326061576429936438942385136577780, 81965165964256317055119897941060305766, 321363256779700398308667095128764232963, 1260314855059917661073398920496915968563, 4943957701328629452439392991174654120405, 19399044264475999654532396424147180129812, 76136511520035119719342376317693598319313, 298889000446021721776388303024708396736146, 1173622913018793109476230285715677154578626, 4609421089573339698174541583179874818260311, 18107600733515458441591906537708518980267989, 71149165104816474866012105793103591020642149, 279621664782415878824955816482026786316841538, 1099162123960505455060348898691658933811994791, 4321559695249450487150610838911451652022682388, 16994370080276012125408459129954834703352388176, 66842619144482121593583783039759695731060810445, 262956423633532088472839987060955326120097525181, 1034651801203892249502567429397055623617724711131, 4071767679803451979663948260886566197048154294178, 16026858091585020383869427046072052037444517713749, 63094102257164560327564621980765327839208381396620, 248429125421833376440333501509650353684120272154540, 978336093958173485287823930732889015790138384713049, 3853398830675151643060600773635521652450297934628125, 15179893002875710820931224192459677690595810466954667, 59808225048365835500195278467214347869764352274567134, 235678087310028615976062327176524786036952969451690861, 928842883597491171906022853324044149959236752927273704, 3661244966226301323467613773328518420215447548062068682, 14433695728294641325057057697035537324040080096929579463, 56909834394863649073830651063659752016766619896327088655, 224417613515778576571077673570523859322791782476358767681, 885085564354580395446870099090797369752749130129620599318, 3491171356851462535675136286724447277719846040957435268343
Generating function in Maple syntax:
1/2*((x^3-2*x^2+2*x-1)*(1-4*x)^(1/2)+2*x^4-3*x^3+12*x^2-6*x+1)/x/(x^3-2*x^2+5*x-1)
Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+2 x -1\right) \sqrt{1-4 x}+2 x^{4}-3 x^{3}+12 x^{2}-6 x +1}{2 x \left(x^{3}-2 x^{2}+5 x -1\right)}
Generating function in sympy syntax:
(x**4 - 3*x**3/2 + 6*x**2 - 3*x + sqrt(1 - 4*x)*(x**3 - 2*x**2 + 2*x - 1)/2 + 1/2)/(x*(x**3 - 2*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^3-2*x^2+5*x-1)*F(x)^2+(-2*x^4+3*x^3-12*x^2+6*x-1)*F(x)+x^4+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{3}-2 x^{2}+5 x -1\right) F \! \left(x \right)^{2}+\left(-2 x^{4}+3 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+x^{4}+5 x^{2}-4 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 68
a(6) = 235
a(7) = 824
a(n+7) = 2*(1+2*n)/(n+8)*a(n)-(26+17*n)/(n+8)*a(1+n)+2*(49+24*n)/(n+8)*a(n+2)-(238+75*n)/(n+8)*a(n+3)+2*(163+36*n)/(n+8)*a(n+4)-2*(125+21*n)/(n+8)*a(n+5)+(78+11*n)/(n+8)*a(n+6), n >= 8
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) = 68
a \! \left(6\right) = 235
a \! \left(7\right) = 824
a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{\left(26+17 n \right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(49+24 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(238+75 n \right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(163+36 n \right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(125+21 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(78+11 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/21227/
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[4,x] = F[12,x]*F[5,x]
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[19,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]+F[16,x]+F[18,x]
F[10,x] = F[11,x]
F[11,x] = F[13,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[13,x]^2*F[12,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[13,x]*F[9,x]
F[18,x] = F[12,x]*F[9,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]*F[3,x]
F[21,x] = F[1,x]+F[22,x]+F[23,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[13,x]*F[21,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_{4}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{19}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{13} \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_{13} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{21}\! \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))
Eq(F_4(x), F_12(x)*F_5(x))
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_19(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x) + F_16(x) + F_18(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(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)**2)
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_13(x)*F_9(x))
Eq(F_18(x), F_12(x)*F_9(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x)*F_3(x))
Eq(F_21(x), F_1(x) + F_22(x) + F_23(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_13(x)*F_21(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"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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_and_row_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": [1, 0, 3, 2], "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], "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], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [1, 0, 3, 2], "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_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"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], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 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, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 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, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [2, 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": 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