0132_0231_0321_1032_2031
Counting sequence:
1, 1, 2, 6, 19, 63, 218, 779, 2852, 10642, 40325, 154752, 600236, 2349320, 9267240, 36804991, 147045409, 590591250, 2383220253, 9657669951, 39285464437, 160357145231, 656610561025, 2696335457970, 11101565519174, 45819315000614, 189533984324786, 785648987932968, 3262941841833322, 13576013964188150, 56580382431247912, 236180239852785125, 987331636404047232, 4133194982201437942, 17325148721223067795, 72711675937223213721, 305518816007108399286, 1285142477958021994362, 5411518013203607289175, 22809636324927946917350, 96233805806953880077387, 406375189254110818086198, 1717506976425048065137250, 7264813831852298047731971, 30753116700884443088158017, 130279930901306091358716369, 552300803197157879665323357, 2342990469040020824044241270, 9946030625031167251912629880, 42247592168783813768290048442, 179562728040067872508360069662, 763628777399636024587138836118, 3249306036100791906923055143044, 13833501094885898912503423791168, 58924808951447703690420007760426, 251119667287776858364677600883526, 1070710496353523441642259459690372, 4567356404514141937379812551420880, 19491818025699651196627169498010802, 83220024948131533467378207659773428, 355455429282333938006559016632080132, 1518862128817092845447774855595716900, 6492647208417859439236864673744144188, 27764517666801107146547789050980895157, 118773133463291338960001134463945399773, 508277821544505173100236576955637224700, 2175876807907384707706895253044835334287, 9317795714100386043286451601813624833243, 39914761677665179140299371875164198135157, 171037480958136293794599089808677844534001, 733132601538519294360773826661330370478133, 3143428033735130515111851409058999908815489, 13481886727719275491237812989515376774106369, 57838969810034154246838942442627975722461684, 248204611602498941296179529376783450656577935, 1065406634199498240542223284599954319203508987, 4574400183371979704328171470469013689683453167, 19645504931191909702452826251143015913975582279, 84391693605516814726643983881869825190036399725, 362610994628530931799372623349359715388248815252, 1558419540185998443861772980113621525171566505471, 6699269393048096614311080719379149953103878826294, 28804989416889449814704470743410589747386479040444, 123880481888050409333192519472172589205087533324928, 532881558429053634877266365968749140792931609835181, 2292709146559728528101767129946308095797225431566491, 9866329366178058217281859603135700036790206643169152, 42466708879694876111503170974977582692976552362510550, 182820964565219023214158393958141640741999931750132783, 787201464558914692666364671818357242251299587998487852, 3390208990119754126234390240357810427504578724647446718, 14603130394091263146205559758469350886059708403847424475, 62913328171614005557916944634968568058027471205034556797, 271090881025679241091899284679457567876421406740948579457, 1168318107290131078875068979717638817424474179269537446597, 5035930240939102702639042598593600574155183213795286656098, 21710468718660948672971808567903857027073726896507167493414, 93611269320048394829462889856895723323910871465898523431226, 403696604469258097554559796885588775787845049249404798424506, 1741200237716604189731453788011025505238163336823280733996806, 7511171348548630920110044858526236245565773553628985153548094
Implicit equation for the generating function in Maple syntax:
-x^2*F(x)^3+x*(x^2+2)*F(x)^2+(-x-1)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
-x^{2} F \! \left(x \right)^{3}+x \left(x^{2}+2\right) F \! \left(x \right)^{2}+\left(-x -1\right) F \! \left(x \right)+1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 63
a(6) = 218
a(7) = 779
a(n+8) = -4/5*(2*n+3)*(n-1)/(n+9)/(n+8)*a(n)+2/25*n*(11*n+31)/(n+9)/(n+8)*a(n+1)-1/25*(259*n^2+1019*n+840)/(n+9)/(n+8)*a(n+2)+1/25*(325*n^2+1841*n+2364)/(n+9)/(n+8)*a(n+3)-2/25*(266*n^2+2089*n+3930)/(n+9)/(n+8)*a(n+4)+1/25*(445*n^2+4793*n+12696)/(n+9)/(n+8)*a(n+5)-4/25*(91*n^2+1175*n+3771)/(n+9)/(n+8)*a(n+6)+1/5*(35*n+249)/(n+9)*a(n+7), 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) = 19
a \! \left(5\right) = 63
a \! \left(6\right) = 218
a \! \left(7\right) = 779
a \! \left(n +8\right) = -\frac{4 \left(2 n +3\right) \left(n -1\right) a \! \left(n \right)}{5 \left(n +9\right) \left(n +8\right)}+\frac{2 n \left(11 n +31\right) a \! \left(n +1\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{\left(259 n^{2}+1019 n +840\right) a \! \left(n +2\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{\left(325 n^{2}+1841 n +2364\right) a \! \left(n +3\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{2 \left(266 n^{2}+2089 n +3930\right) a \! \left(n +4\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{\left(445 n^{2}+4793 n +12696\right) a \! \left(n +5\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{4 \left(91 n^{2}+1175 n +3771\right) a \! \left(n +6\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{\left(35 n +249\right) a \! \left(n +7\right)}{5 n +45}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21671/
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[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[10,x]+F[9,x]
F[9,x] = F[0,x]*F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[0,x]*F[12,x]*F[4,x]
F[12,x] = F[0,x]+F[10,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
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_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{0}\! \left(x \right) F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{0}\! \left(x \right)+F_{10}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_10(x) + F_9(x))
Eq(F_9(x), F_0(x)*F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_0(x)*F_12(x)*F_4(x))
Eq(F_12(x), F_0(x) + F_10(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": [0, 3, 2, 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": [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": [0, 3, 2, 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": [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": [0, 3, 2, 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": [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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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, 3, 2, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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, 3, 2, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "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, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 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