0132_0213_1032_1320
Counting sequence:
1, 1, 2, 6, 20, 67, 224, 752, 2547, 8722, 30211, 105793, 374189, 1335403, 4803709, 17401219, 63425844, 232450459, 856072404, 3166527634, 11758662714, 43819827604, 163824995274, 614276361154, 2309481770472, 8704391092654, 32881560488776, 124475250270804, 472133202027164, 1794065379929976, 6828909334697148, 26034911372848398, 99405256226653691, 380077218391909170, 1455150132236290363, 5578077502700193497, 21407802715633670153, 82251327511922311819, 316352775612276674585, 1217965207650381834153, 4693652789354170935635, 18104219579424924546229, 69891246766115501709107, 270036553265525046470903, 1044150141806433270126335, 4040442114415462029994019, 15646105862105021317001567, 60629204750081759753239403, 235094965836691605033762965, 912176778262764168173044043, 3541424516307905146776408917, 13757197547148845899982617889, 53471857165889272047375970385, 207947988730370665235265383189, 809113970674657420018152412913, 3149793074829898785239354161985, 12267722977308288279716428177037, 47802232599134435219813158404665, 186349286301912669449141839469645, 726768375008346070966764744573605, 2835611784887495335591295020061429, 11068124275246581463088113596955181, 43218740789007281129171867922680117, 168824555478251121767952729794452511, 659719606939702795617477973317127172, 2578923883202087096398281185933766107, 10084823931362988122405417677206725252, 39449753821856256978807806163916038242, 154369738965839013376558916213038381922, 604251639850649786710845191168042063732, 2365959883860089528081756473304948953202, 9266741055801755779403165098758996617042, 36305563963012909009844209575087462169172, 142279523143726628758082422995435674332382, 557739435375138759122257819231092905985332, 2186939955686829640570904679944857933848392, 8577403934836813413445191567105696665392592, 33650027536914971091981573203375339069319332, 132045229734579759909425053968352555099474352, 518280681357017815350199787319975168904945412, 2034743734823629498557647787529563698482468962, 7990138220717461764097893014280122969574937972, 31383133838276525591473121714072801929463666722, 123291583001608511924164755763124129508267865702, 484466100536930475052299328524786843777596367062, 1904074887090200833174625770835650099182261424162, 7485024402996859798130022896484367633589084092342, 29429911058661879914358850415126389424777703569542, 115736208510833160613955599838515468679223036077082, 455231419691938377312548836737122946667582079768622, 1790919444792527772697267043597790485858793899299162, 7046913604101475122093793297055776881395203337677442, 27733149979913598514200366035161940971528342967567602, 109162922079063128967435925685392069015692322817590522, 429759214089992511115981088340683281987117321356854962, 1692184753365059675788982336298837039572237213711815142, 6664098196976303941725432694572568108452871149044964952, 26248504605940183098484320631340469606527836520016013422, 103403656703027881408353840713562542959857861978589002712, 407412178286953546549972331798304385912737334684440390372, 1605452877206542351613109697755810320544311711385244174852
Generating function in Maple syntax:
((x^3-7*x^2+7*x-2)*(1-4*x)^(1/2)-7*x^3+13*x^2-9*x+2)/(4*x^3-6*x^2+2*x)
Generating function in latex syntax:
\frac{\left(x^{3}-7 x^{2}+7 x -2\right) \sqrt{1-4 x}-7 x^{3}+13 x^{2}-9 x +2}{4 x^{3}-6 x^{2}+2 x}
Generating function in sympy syntax:
(-7*x**3 + 13*x**2 - 9*x + sqrt(1 - 4*x)*(x**3 - 7*x**2 + 7*x - 2) + 2)/(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+(2*x-1)*(x-1)*(7*x^3-13*x^2+9*x-2)*F(x)+x^6-2*x^5+21*x^4-44*x^3+37*x^2-14*x+2 = 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(2 x -1\right) \left(x -1\right) \left(7 x^{3}-13 x^{2}+9 x -2\right) F \! \left(x \right)+x^{6}-2 x^{5}+21 x^{4}-44 x^{3}+37 x^{2}-14 x +2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 224
a(n+6) = -2*(-1+2*n)/(n+7)*a(n)+5*(8+7*n)/(n+7)*a(1+n)-1/2*(365+161*n)/(n+7)*a(n+2)+(279+82*n)/(n+7)*a(n+3)-6*(32+7*n)/(n+7)*a(n+4)+1/2*(121+21*n)/(n+7)*a(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) = 67
a \! \left(6\right) = 224
a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +7}+\frac{5 \left(8+7 n \right) a \! \left(1+n \right)}{n +7}-\frac{\left(365+161 n \right) a \! \left(n +2\right)}{2 \left(n +7\right)}+\frac{\left(279+82 n \right) a \! \left(n +3\right)}{n +7}-\frac{6 \left(32+7 n \right) a \! \left(n +4\right)}{n +7}+\frac{\left(121+21 n \right) a \! \left(n +5\right)}{14+2 n}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21183/
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[11,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[7,x]^2*F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[4,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[14,x]*F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]*F[16,x]*F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[16,x]*F[22,x]
F[22,x] = F[1,x]+F[20,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_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
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_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \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_11(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x)**2)
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_4(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_14(x)*F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)*F_16(x)*F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_16(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(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, 1, 3], "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, 2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 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"}}, {"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], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 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, 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], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 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": {"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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"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": 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