0132_1302_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 214, 693, 2271, 7575, 25732, 88853, 311091, 1101821, 3940194, 14205586, 51573828, 188377066, 691724786, 2552007934, 9454864406, 35161627428, 131209754066, 491147425176, 1843695818908, 6938988913842, 26178386998326, 98980581264612, 375013621123756, 1423544570135114, 5413344199247322, 20619586462313117, 78662561578829619, 300530403667073191, 1149749092389466224, 4404306367103269409, 16891963994753485435, 64860749394047402665, 249319534954415144246, 959353389408107756607, 3695099896934961098027, 14245510221713160235515, 54968590791741116119720, 212284293279302953886331, 820485541070136768654697, 3173643190463494704515867, 12284703605839048155110170, 47585754336399739450129625, 184451446036544255908108641, 715431301085353467127502969, 2776663458805725578600692800, 10782943491305244634343376829, 41898661878189653433692546163, 162892936729072200331804233721, 633628490651490180230209696742, 2465977123145286647431753077805, 9601902865812581955154325884157, 37405195673322184481189504770689, 145782618002675271416740497859316, 568423405141407649295894662940817, 2217298152457811122248076763179771, 8652787505499403974021404401941741, 33780178870893124490674529524333282, 131927780860655577220528355924670146, 515434733515440967304241050910494596, 2014508792075689129681739702731713770, 7876221086516843606106424550896424706, 30804578734542686203325971014153835142, 120519382540628670269179784450330846766, 471670338888454919919809799523380510316, 1846529527048216760269505669713412872530, 7231129133206016191052022091290090936796, 28325948284579885208143118278363210216952, 110990994078275350738096615954613109183286, 435023354996397326910164911170245376756926, 1705515607113351709793376960324048137221800, 6688272928509021834711474636742214985595016, 26235203396274400313700684633677777719443222, 102935257309693196226838460286953055178992106, 403971163489146296455049729235201781330602906, 1585770716011153536827111213597194832733080910, 6226323268624341118202993840072306201235221016, 24452411774571159444582983791771168334585930002, 96052375422988884571039721625244493464829582126, 377388436090919656961829576635952585724914916626, 1483067488857834683091982342399332450137835854240, 5829387746711093750118041056398804873248041257786, 22917783145400730746640716206498383785850626655622, 90117243923112987744094008368932328526605705595246, 354427081368742440617582881322626659562778929052576, 1394208514416824800711241139504962864852499527657530, 5485403668096838508570718730843571290300187258291226, 21585779591579201701404629600364002577589481235451382, 84957843102651094102752731100876475093654051137638316, 334436890398631121572067847357514013231841000792774746, 1316734608706056329569049033822408015879794896521312640, 5185064532279364541066195605832547920141817209782374916, 20421161332216990482919090712609034736218336545503468842, 80440659294047863509811996195582467502320168372805530286, 316911737357114915602736246007714828495776480740779974756, 1248726103377008599790957724813130739548691110306910818060
Generating function in Maple syntax:
1/2*((-3*x^2+3*x-1)*(1-4*x)^(1/2)-4*x^4+4*x^3+x^2-3*x+1)/x/(x-1)^4
Generating function in latex syntax:
\frac{\left(-3 x^{2}+3 x -1\right) \sqrt{1-4 x}-4 x^{4}+4 x^{3}+x^{2}-3 x +1}{2 x \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**4 + 2*x**3 + x**2/2 - 3*x/2 + sqrt(1 - 4*x)*(-3*x**2 + 3*x - 1)/2 + 1/2)/(x*(x - 1)**4)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^8*F(x)^2+(2*x-1)*(2*x^3-x^2-x+1)*(x-1)^4*F(x)+4*x^7-8*x^6+2*x^5+17*x^4-28*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{8} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(2 x^{3}-x^{2}-x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{7}-8 x^{6}+2 x^{5}+17 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 214
a(n+4) = -6*(2*n+5)/(n+5)*a(n)+3*(9*n+20)/(n+5)*a(n+1)-(22*n+65)/(n+5)*a(n+2)+(31+8*n)/(n+5)*a(n+3)+(n-2)*(n-7)/(n+5), n >= 7
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) = 66
a \! \left(6\right) = 214
a \! \left(n +4\right) = -\frac{6 \left(2 n +5\right) a \! \left(n \right)}{n +5}+\frac{3 \left(9 n +20\right) a \! \left(n +1\right)}{n +5}-\frac{\left(22 n +65\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(n -2\right) \left(n -7\right)}{n +5}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21291/
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[15,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[5,x]*F[8,x]*F[9,x]
F[15,x] = F[16,x]
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[26,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]*F[24,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[5,x]^3
F[23,x] = F[9,x]^2*F[6,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[12,x]*F[24,x]
F[26,x] = F[6,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{15}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \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_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right) F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{5} \left(x \right)^{3}
F_{23}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{26}\! \left(x \right) = F_{6}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_15(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
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))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_5(x)*F_8(x)*F_9(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_26(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x)*F_24(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_5(x)**3)
Eq(F_23(x), F_6(x)*F_9(x)**2)
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_12(x)*F_24(x))
Eq(F_26(x), F_6(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": [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": [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": [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": [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": [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": [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"}}, {"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": [2, 1, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "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], [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": [2, 1, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "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}}, {"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": [2, 1, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], 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