0123_0213_0231_0312_1203_2013
Counting sequence:
1, 1, 2, 6, 18, 54, 167, 534, 1755, 5896, 20167, 70014, 246105, 874175, 3132871, 11313875, 41130823, 150401874, 552807576, 2041191004, 7567894379, 28162561295, 105153557439, 393822147183, 1479064860027, 5569111431208, 21018839617096, 79502037535421, 301318337832209, 1144166027284692, 4352245879014485, 16582456290362267, 63277644601249880, 241811453721078870, 925318690539738102, 3545355022189914316, 13600377289570471013, 52231977496247594818, 200812159993999752581, 772836026864507176426, 2977187220624337817815, 11479570696961623861123, 44302336935836480689371, 171116244849711451327888, 661458835006528992183937, 2558856052308400868820055, 9906171479077144086031308, 38376840024198733848778530, 148772702747064527923894822, 577106592752442713605851669, 2240042981444014561750289499, 8699887887384786774771489641, 33807886115575786127918793113, 131449841595452978518322991939, 511364957841108499528584270612, 1990317443756000066506835479144, 7750439231821940669859618108713, 30195023582466643248295126906675, 117690774897358232187670841002316, 458923953199868691252313721036203, 1790292059355152280346741790724022, 6986920764650283218724982958538760, 27278515820704381053417251572685637, 106542561278220757651545492410290897, 416282116814721021807568067569616594, 1627082574436521166830739849311740531, 6361856562183762514421465365176640400, 24883188388037471764583609467930160471, 97358011624594126328941019160356894899, 381045384446765556660970765520003616797, 1491822197735644342862994997966601615204, 5842368570448141783649076973145677328329, 22886991129224606693530632146581000348019, 89683552471449911054877092305363700612040, 351526351595012197745583905469471023070566, 1378226889825751105185716357294058697905508, 5405031053138203591677708126981189707676533, 21202512383544464523741331749359799253645325, 83192705042729224136530859376969298014743331, 326504517030847505146333233283426329936272209, 1281729893725602268328096928496917416214840303, 5032741762311736634460969177722401610151270296, 19765651304555071139086844560878324122734844402, 77645016628699006772016885438571798613590097293, 305077184979582351703988690017024997120829292245, 1198939725039114658215033721389158312780865767996, 4712749055341929248047351145282183550261166323809, 18528427709774923930825026304874223750710765844464, 72859824323630235156166128493661066449797312427030, 286563618048988471837226109995014573870071795340653, 1127289618106137344374312244723283980097260488293354, 4435368334469729665534463145066513095524129436731542, 17454277176064193045317909600649763377808711124896040, 68698990845776236211454514639148442139553818468547928, 270441622531805465124553679775794293942101738323180743, 1064804356693142056929826344027671782591802151606777108, 4193123864756457456422131335305754389902687067178108331, 16514889464281025815196013431992203931923291405948935030, 65055246281009470377583499989350162482307090572628574553, 256304519542568843702840770536672922903710867433435919771, 1009941319447806829867422254014251085498215665305792959131
Generating function in Maple syntax:
-(1/2*(x-1)^2*(-4*x+1)^(1/2)+(x^2+1)*(x-1/2))*(x-1)^2/x/(x^5-2*x^3+5*x^2-4*x+1)
Generating function in latex syntax:
-\frac{\left(\frac{\left(x -1\right)^{2} \sqrt{-4 x +1}}{2}+\left(x^{2}+1\right) \left(x -\frac{1}{2}\right)\right) \left(x -1\right)^{2}}{x \left(x^{5}-2 x^{3}+5 x^{2}-4 x +1\right)}
Generating function in sympy syntax:
(x - 1)**2*(-sqrt(1 - 4*x)*(x - 1)**2/2 - (x - 1/2)*(x**2 + 1))/(x*(x**5 - 2*x**3 + 5*x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^5-2*x^3+5*x^2-4*x+1)*F(x)^2+(2*x-1)*(x^2+1)*(x-1)^2*F(x)+(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{5}-2 x^{3}+5 x^{2}-4 x +1\right) F \! \left(x \right)^{2}+\left(2 x -1\right) \left(x^{2}+1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 54
a(6) = 167
a(n+7) = -2*(2*n+3)/(8+n)*a(n)+(5*n+29)/(8+n)*a(1+n)+(7*n+4)/(8+n)*a(n+2)-2*(15*n+44)/(8+n)*a(n+3)+(43*n+185)/(8+n)*a(n+4)-(162+29*n)/(8+n)*a(n+5)+(9*n+61)/(8+n)*a(n+6), 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) = 18
a \! \left(5\right) = 54
a \! \left(6\right) = 167
a \! \left(n +7\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{8+n}+\frac{\left(5 n +29\right) a \! \left(1+n \right)}{8+n}+\frac{\left(7 n +4\right) a \! \left(n +2\right)}{8+n}-\frac{2 \left(15 n +44\right) a \! \left(n +3\right)}{8+n}+\frac{\left(43 n +185\right) a \! \left(n +4\right)}{8+n}-\frac{\left(162+29 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(9 n +61\right) a \! \left(n +6\right)}{8+n}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22098/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[14,x]*F[6,x]
F[6,x] = F[15,x]+F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[12,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[0,x]+F[10,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]*F[8,x,k[0]]
F[11,x,k[0]] = k[0]*x
F[12,x,k[0]] = F[13,x,k[0]]*F[14,x]
F[13,x,k[0]] = (F[8,x,k[0]]*k[0]-F[8,x,1])/(-1+k[0])
F[14,x] = x
F[15,x] = F[16,x]
F[16,x] = F[17,x]^2*F[0,x]*F[14,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[14,x]*F[17,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{14}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{15}\! \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_{12}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = y x
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{13}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17} \left(x \right)^{2} F_{0}\! \left(x \right) F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_14(x)*F_6(x))
Eq(F_6(x), F_15(x) + F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_12(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_0(x) + F_10(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0)*F_8(x, k_0))
Eq(F_11(x, k_0), k_0*x)
Eq(F_12(x, k_0), F_13(x, k_0)*F_14(x))
Eq(F_13(x, k_0), (-k_0*F_8(x, k_0) + F_8(x, 1))/(1 - k_0))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_0(x)*F_14(x)*F_17(x)**2)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_14(x)*F_17(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_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, 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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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], "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], [2, 1]]}, {"patt": [0, 1, 2], "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]]}, {"patt": [2, 0, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 1, 2, 3], "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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "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]]}, {"patt": [2, 0, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 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], [2, 1]]}, {"patt": [0, 1, 2], "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]]}, {"patt": [2, 0, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], 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