0123_0132_0213_0231_1203_1230_2013
Counting sequence:
1, 1, 2, 6, 17, 51, 160, 519, 1727, 5863, 20228, 70720, 250002, 892126, 3209328, 11626385, 42378075, 155307615, 571925820, 2115257100, 7853744910, 29263124250, 109384710240, 410075910270, 1541481197334, 5808790935126, 21939395326280, 83038957586704, 314914149594212, 1196456617485724, 4553480766993312, 17357380324667845, 66263714414182803, 253325559250064399, 969746107684925676, 3716893406644788916, 14263139838618023990, 54794317215574894594, 210724868725767990272, 811208430192360759426, 3125818551219461207810, 12055620262184414442450, 46536221461265849984440, 179783990525351897485920, 695109168886177600789500, 2689564494465159414681060, 10414146446017424770212000, 40351984614646972203222090, 156456343287827238974375070, 607011369664595456646170982, 2356486004693608426200391128, 9153495083090859900799982040, 35575690958530781681749459212, 138342252731040743815465447236, 538248422537963279748020937792, 2095216463180989908189535539516, 8159911826425738897094528830860, 31793995168010906530802399501004, 123936956218052282440193881260432, 483332502559924437859896541266528, 1885707499434574724383917903593928, 7360033928053398707443210182888120, 28738014737034362433511766614450752, 112253460343232750963159762645437101, 438635282691543012441232969113653763, 1714601892467964348563563552011032175, 6704620987835415744387396557379463500, 26225988033365910529479978425662229700, 102619967436356490454550277571908913350, 401670701487219743819610881504702783250, 1572688686931537960320540329981587060800, 6159506530556171854352098577730236636250, 24131038205935223151857996585059477197450, 94564809830140672144718296006608844193850, 370683477286471299115815530582903761886040, 1453429043376404755125088775894421995933760, 5700307706482579402019610638884017457788300, 22362156002742809674710986765474447523483540, 87747977407979039123573126925340009971561120, 344402182059987473338762636605050551310477790, 1352064415511068712926226571179510894692670570, 5309199546760391854940590328481309872902876450, 20852515487657761045266748938417121363228275080, 81918736098408651060159951255078169595881785640, 321885307156558535901795374805921557927260252260, 1265056712828888759293791069297976378000789478860, 4972876325439403452459090022925215350731183438400, 19552041205299677695610804130890317143073238304740, 76888495346871816433051086618738160147620374341620, 302422104055098284097380933304224942405861316892660, 1189725498518592664082856053829237329986326926716400, 4681222676080358670823299382521049727049945397948000, 18422535455275539310616781286005150877425501717908600, 72512922124248300473173036971844642614209628355249800, 285466864230925531513250013240528028666236625267832000, 1124006266960164797270059083497586670951541856815695650, 4426423656865305588247560635210709533540456628125368750, 17434398762558101614728142517581339824585230890043545350, 68679844221324978833640707913320642436024179878277671800, 270594259507701132006990363157074677528349459275397903240, 1066285419814759488566801909861746630042858472515530345660
Generating function in Maple syntax:
1/2*((x^2-x-1)*(1-4*x)^(1/2)-3*x^2+x+1)/x
Generating function in latex syntax:
\frac{\left(x^{2}-x -1\right) \sqrt{1-4 x}-3 x^{2}+x +1}{2 x}
Generating function in sympy syntax:
(-3*x**2/2 + x/2 + sqrt(1 - 4*x)*(x**2 - x - 1)/2 + 1/2)/x
Implicit equation for the generating function in Maple syntax:
x*F(x)^2+(3*x^2-x-1)*F(x)+x^4-2*x^2+x+1 = 0
Implicit equation for the generating function in latex syntax:
x F \! \left(x \right)^{2}+\left(3 x^{2}-x -1\right) F \! \left(x \right)+x^{4}-2 x^{2}+x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(n+3) = -2*(-3+2*n)/(4+n)*a(n)+(2+5*n)/(4+n)*a(1+n)+(8+3*n)/(4+n)*a(n+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) = 17
a \! \left(n +3\right) = -\frac{2 \left(-3+2 n \right) a \! \left(n \right)}{4+n}+\frac{\left(2+5 n \right) a \! \left(1+n \right)}{4+n}+\frac{\left(8+3 n \right) a \! \left(n +2\right)}{4+n}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/22709/
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[13,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[1,x]+F[11,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[8,x,k[0]]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[12,x,k[0]]*F[13,x]
F[12,x,k[0]] = (F[8,x,k[0]]*k[0]-F[8,x,1])/(-1+k[0])
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[24,x]
F[16,x] = F[17,x,1]
F[17,x,k[0]] = F[18,x,k[0]]+F[21,x,k[0]]
F[18,x,k[0]] = F[19,x,k[0]]+F[5,x]
F[19,x,k[0]] = F[20,x,k[0]]
F[20,x,k[0]] = F[10,x,k[0]]*F[17,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[13,x]*F[23,x,k[0]]
F[23,x,k[0]] = (F[17,x,k[0]]*k[0]-F[17,x,1])/(-1+k[0])
F[24,x] = F[1,x]+F[13,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x , 1\right)
F_{8}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{13}\! \left(x \right)
F_{12}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{24}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x , 1\right)
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{5}\! \left(x \right)
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{13}\! \left(x \right) F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = \frac{y F_{17}\! \left(x , y\right)-F_{17}\! \left(x , 1\right)}{-1+y}
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_14(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_1(x) + F_11(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_8(x, k_0))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_12(x, k_0)*F_13(x))
Eq(F_12(x, k_0), (-k_0*F_8(x, k_0) + F_8(x, 1))/(1 - k_0))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_24(x))
Eq(F_16(x), F_17(x, 1))
Eq(F_17(x, k_0), F_18(x, k_0) + F_21(x, k_0))
Eq(F_18(x, k_0), F_19(x, k_0) + F_5(x))
Eq(F_19(x, k_0), F_20(x, k_0))
Eq(F_20(x, k_0), F_10(x, k_0)*F_17(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_13(x)*F_23(x, k_0))
Eq(F_23(x, k_0), (-k_0*F_17(x, k_0) + F_17(x, 1))/(1 - k_0))
Eq(F_24(x), F_1(x) + F_13(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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[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, 1, 2], "pos": [[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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 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