0123_0231_1230
Counting sequence:
1, 1, 2, 6, 21, 78, 297, 1144, 4433, 17238, 67184, 262276, 1025202, 4011660, 15712335, 61590780, 241610745, 948453990, 3725581860, 14643037860, 57585238230, 226578907620, 891954995490, 3512930443776, 13841663734746, 54561842949628, 215160421702352, 848788862905704, 3349598987229572, 13223131180980952, 52217627747388411, 206269383829036164, 815044069813148553, 3221442435682888326, 12736123327012448636, 50365911512285571748, 199225062585517554222, 788233028030049888244, 3119352678190668279710, 12347231135782975499760, 48883939859583174314430, 193575697695767213123220, 766690623724669081381440, 3037183381329051746768760, 12033735075951170907403260, 47687617640398028441296680, 189009608386904874559751670, 749260219444797853569174840, 2970637058237871047112176778, 11779648962093502073850656988, 46717409917780101589422981384, 185304831828628232836320434856, 735113386498737405328505053212, 2916618554189737121818452021096, 11573397065215808536712000652036, 45929981570419692632224737814848, 182298983442493789771019362248852, 723641078270898104998103529480504, 2872841503979775591324553599213408, 11406388879486343500693817408708688, 45293028390789045736833034471547784, 179870484890254609640362209411865776, 714385363833256518873398476353775827, 2837579749122430653310794045463377300, 11272112787564755083153322079373504425, 44781968475721875675543230592686142150, 177926471229934584628925454943587701100, 706995176791474342954722253442349089700, 2809505223451834453448992090770031835550, 11165541484457708825111058180646939569300, 44377759141114479583953891450638542206150, 176394766746842611394990728803557856195600, 701196790638734516881558207245454410604950, 2787579206926914414669584976142812352468260, 11082731038858202373103922792984998663284960, 44065417246851848561822042671923620770616280, 175218418952903923833588280160904465478243020, 696773608187672809123975660587942023683916040, 2770977528703332126622857455938373396168035170, 11020543831413789505148288656233944768436859720, 43832992012773471074631155975192233724873937710, 174351887133881003911184235274848071254088919540, 693552007046549597603090560494662373040250985080, 2759037958802933449118431826236251907870677787960, 10976452868330808916301552958324524746996768575060, 43670836557945797884271953783785533184110901889080, 173758328512156448623349515221013403215140594406620, 691391221808846718294684030708946581132227622906240, 2751222458933974721704140202326955790928144017609900, 10948400839487682654696412466069015548705265395299400, 43571081304432847456228610827722389047442870092639200, 173407629389540335492928644528862433237139589513306800, 690175977855516220494261294162419997860705930792588600, 2747089604713603366652749838837176072700235888976232400, 10934696805167383548008725072407158191973670202432461150, 43527246694465456007066422981367733182152031214366508200, 173274952466371121033958584492994923553617133266724307050, 689811039093006989505673069571465814513339970871130899100, 2746274109790240576120847040918518247246025996073424419160, 10933939254209794685727446030501719773293329973741388552360, 43533953901347267677368170167421768944905963178249102831980
Generating function in Maple syntax:
(-9*(x-1/3)^2*(-4*x+1)^(1/2)-4*x^3+17*x^2-8*x+1)/(8*x^3-2*x^2)
Generating function in latex syntax:
\frac{-9 \left(x -\frac{1}{3}\right)^{2} \sqrt{-4 x +1}-4 x^{3}+17 x^{2}-8 x +1}{8 x^{3}-2 x^{2}}
Generating function in sympy syntax:
(-4*x**3 + 17*x**2 - 8*x - 9*sqrt(1 - 4*x)*(x - 1/3)**2 + 1)/(8*x**3 - 2*x**2)
Implicit equation for the generating function in Maple syntax:
x^2*(4*x-1)*F(x)^2+(4*x-1)*(x^2-4*x+1)*F(x)+x^3+12*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(4 x -1\right) F \! \left(x \right)^{2}+\left(4 x -1\right) \left(x^{2}-4 x +1\right) F \! \left(x \right)+x^{3}+12 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(2+n) = -6*(1+2*n)/(4+n)*a(n)+(17+7*n)/(4+n)*a(n+1), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(2+n \right) = -\frac{6 \left(1+2 n \right) a \! \left(n \right)}{4+n}+\frac{\left(17+7 n \right) a \! \left(n +1\right)}{4+n}, \quad n \geq 3
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/239/
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[9,x]
F[3,x] = F[1,x]+F[25,x]+F[4,x]
F[4,x] = F[5,x]*F[9,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[23,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]*F[9,x]
F[8,x,k[0]] = 1/(-1+k[0])*(F[6,x,k[0]]*k[0]-F[6,x,1])
F[9,x] = x
F[10,x,k[0]] = F[11,x,k[0]]*F[22,x,k[0]]
F[11,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[12,x,k[0]]+F[14,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[9,x]
F[13,x,k[0]] = 1/(-1+k[0])*(F[11,x,k[0]]*k[0]-F[11,x,1])
F[14,x,k[0]] = F[15,x,k[0]]*F[9,x]
F[15,x,k[0]] = F[1,x]+F[14,x,k[0]]+F[16,x,k[0]]+F[21,x,k[0]]
F[16,x,k[0]] = F[17,x,k[0]]*F[9,x]
F[17,x,k[0]] = (F[18,x,k[0]]*k[0]-F[18,x,1])/(-1+k[0])
F[18,x,k[0]] = 1/(-1+k[0])*(F[19,x,k[0]]*k[0]-F[19,x,1])
F[19,x,k[0]] = F[1,x]+F[20,x,k[0]]+F[21,x,k[0]]
F[20,x,k[0]] = F[18,x,k[0]]*F[9,x]
F[21,x,k[0]] = F[19,x,k[0]]*F[22,x,k[0]]
F[22,x,k[0]] = k[0]*x
F[23,x,k[0]] = F[24,x,k[0]]*F[9,x]
F[24,x,k[0]] = 1/(-1+k[0])*(F[11,x,k[0]]*k[0]-F[11,x,1])
F[25,x] = F[10,x,1]
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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{8}\! \left(x , y\right) = -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}
F_{9}\! \left(x \right) = x
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{13}\! \left(x , y\right) = -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{17}\! \left(x , y\right) = -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}
F_{18}\! \left(x , y\right) = -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}
F_{19}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{21}\! \left(x , y\right) = F_{19}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = y x
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{24}\! \left(x , y\right) = -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}
F_{25}\! \left(x \right) = F_{10}\! \left(x , 1\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)*F_9(x))
Eq(F_3(x), F_1(x) + F_25(x) + F_4(x))
Eq(F_4(x), F_5(x)*F_9(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_10(x, k_0) + F_23(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0)*F_9(x))
Eq(F_8(x, k_0), (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_9(x), x)
Eq(F_10(x, k_0), F_11(x, k_0)*F_22(x, k_0))
Eq(F_11(x, k_0), F_1(x) + F_10(x, k_0) + F_12(x, k_0) + F_14(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_9(x))
Eq(F_13(x, k_0), (k_0*F_11(x, k_0) - F_11(x, 1))/(k_0 - 1))
Eq(F_14(x, k_0), F_15(x, k_0)*F_9(x))
Eq(F_15(x, k_0), F_1(x) + F_14(x, k_0) + F_16(x, k_0) + F_21(x, k_0))
Eq(F_16(x, k_0), F_17(x, k_0)*F_9(x))
Eq(F_17(x, k_0), (-k_0*F_18(x, k_0) + F_18(x, 1))/(1 - k_0))
Eq(F_18(x, k_0), (k_0*F_19(x, k_0) - F_19(x, 1))/(k_0 - 1))
Eq(F_19(x, k_0), F_1(x) + F_20(x, k_0) + F_21(x, k_0))
Eq(F_20(x, k_0), F_18(x, k_0)*F_9(x))
Eq(F_21(x, k_0), F_19(x, k_0)*F_22(x, k_0))
Eq(F_22(x, k_0), k_0*x)
Eq(F_23(x, k_0), F_24(x, k_0)*F_9(x))
Eq(F_24(x, k_0), (-k_0*F_11(x, k_0) + F_11(x, 1))/(1 - k_0))
Eq(F_25(x), F_10(x, 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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_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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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], "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": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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"}}, {"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, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 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": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [0, 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", 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