0132_0231_1302_1320_2031
Counting sequence:
1, 1, 2, 6, 19, 60, 190, 609, 1986, 6599, 22325, 76761, 267675, 944725, 3368601, 12116578, 43908987, 160150120, 587409726, 2165210398, 8016058692, 29793450610, 111124358286, 415796897428, 1560314973006, 5870763924592, 22142888137532, 83704467467316, 317076984345688, 1203421032250732, 4575612835450100, 17426399624752937, 66473089469002994, 253934313056282351, 971394605774987093, 3720776950907234877, 14269315668759472443, 54786658756226233249, 210582189901882589645, 810249448536016225719, 3120636481251868309717, 12030224126681921686443, 46418454681362254323535, 179256861035008633782759, 692806645455755362684867, 2679684147210762869460783, 10372314658801197132561895, 40176716866805336594815809, 155728084388009360508166027, 604005691155099590614787805, 2344149521633097386721955801, 9103094615373778817459610297, 35370579710353300434256211701, 137510277530319849223668456321, 534883277668176993348868169881, 2081638354836197041000580482341, 8105240741274514435219052471929, 31574271437217837248229681598749, 123055305331436851687662571319413, 479799845950409509909412500429605, 1871570287269110823948249764023789, 7303521220085739931644617131413141, 28512329846798418149347886207932837, 111352968305613440847016676210181426, 435045076651881086834691519727905115, 1700297908544202008609310127705717368, 6647666947063847558972020494919531774, 25999341011831476924691716408588271286, 101718482828138407197222838175003444260, 398086672039956938510587611013959939258, 1558445423350109202135464875508029925254, 6102923032886320404004684447285722853896, 23906325192990672803885128108681241221630, 93672657483086872534324077941042820913548, 367142403448278659197900551932291230748224, 1439377382970963214464830060263370585780376, 5644559944021809764253401993852934766577860, 22141027965134612140543030680958901330722368, 86871006229552933361163514697740121501670424, 340924741279486753603251957034113976969702086, 1338277276635493343035280636186998830498209460, 5254543907019612886930532199073371652845213818, 20635870767281676177861350581257375753439334414, 81060080723700628410795341719065463738923043086, 318482383324443397285847052733591231610794825410, 1251571661728358001046723899042619468244583571558, 4919441536444075789249990835568659723033616005934, 19340317008638185467100692159001333376862327540226, 76049624255462613005635384894236926092384768627950, 299098559799819801886375733567399872652035188231818, 1176558338534175315116885271572181233262199607093554, 4629058802097829070779420491071992812428818971068226, 18215884653366076278767454927169536266541277056634170, 71694276124519332281074667207149290900016223791768658, 282223848446357270421602158539431606193379551411460034, 1111159384119251586302903372530994611274403669143372036, 4375532302492940995578306060398978659653108809745866990, 17232799276084316718012195270019058329506646224419241248, 67881232837697562790708652381468715697224107917959418364, 267430644614738057562408638715899413671777383885010252236, 1053752997085066568154083162095923449175658604427504520280
Generating function in Maple syntax:
((-3*x^2+3*x-1)*(1-4*x)^(1/2)-2*x^3+3*x^2-3*x+1)/(4*x^3-6*x^2+2*x)
Generating function in latex syntax:
\frac{\left(-3 x^{2}+3 x -1\right) \sqrt{1-4 x}-2 x^{3}+3 x^{2}-3 x +1}{4 x^{3}-6 x^{2}+2 x}
Generating function in sympy syntax:
(-2*x**3 + 3*x**2 - 3*x + sqrt(1 - 4*x)*(-3*x**2 + 3*x - 1) + 1)/(4*x**3 - 6*x**2 + 2*x)
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^2*F(x)^2+(x-1)*(x^2-x+1)*(2*x-1)^2*F(x)+x^5+6*x^4-15*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)^{2}+\left(x -1\right) \left(x^{2}-x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}+6 x^{4}-15 x^{3}+14 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) = 60
a(n+5) = 12*(1+2*n)/(n+6)*a(n)-6*(15+11*n)/(n+6)*a(1+n)+(174+71*n)/(n+6)*a(n+2)-2*(69+19*n)/(n+6)*a(n+3)+2*(24+5*n)/(n+6)*a(n+4), 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) = 19
a \! \left(5\right) = 60
a \! \left(n +5\right) = \frac{12 \left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{6 \left(15+11 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(174+71 n \right) a \! \left(n +2\right)}{n +6}-\frac{2 \left(69+19 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21722/
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[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[5,x]*F[8,x]
F[11,x] = F[12,x]+F[4,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[15,x]*F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[14,x]*F[17,x]*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_{5}\! \left(x \right)+F_{9}\! \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_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \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_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_5(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_4(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_15(x)*F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_14(x)*F_17(x)*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.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": [1, 3, 0, 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": [2, 0, 3, 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": [1, 3, 0, 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": [2, 0, 3, 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": [1, 3, 0, 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": [2, 0, 3, 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"}}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 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, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "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, 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], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 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}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 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, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[1, 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]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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