0132_0321_1302
Counting sequence:
1, 1, 2, 6, 21, 76, 279, 1043, 3979, 15464, 61035, 243956, 985332, 4015149, 16486978, 68152082, 283379950, 1184463394, 4973854781, 20973790058, 88776205172, 377051017255, 1606407682084, 6863546019199, 29402042385724, 126256434419874, 543372408355499, 2343361348375852, 10125478262339939, 43829833290592250, 190042630979757845, 825302202679253125, 3589333427536353420, 15632025731596395820, 68168309746458287269, 297634787965910601787, 1301036478838790300085, 5693416223419864777512, 24940734199244300834785, 109364260254309080646902, 480009950733904135541911, 2108696038937207369374987, 9271464145652305159399126, 40797749389997924595724731, 179664272697623514267539742, 791790012518021343681103269, 3491943393330160825385542244, 15410624226475101737422489641, 68054510275760100724555332178, 300722393428812916507019779613, 1329645929076771820767595303654, 5882439489059396639081457953246, 26038776213256446247282580822524, 115323089197848902435716085449375, 511017789053890711364250411475170, 2265540029825848482562717211043707, 10048836473870886944858453490891585, 44592403554629165294493832814241770, 197970357449765380669357282919957989, 879279811378200290588586612568868498, 3906928608815300779833935727509477696, 17366783170480064420727228710779439949, 77227717399803033058588303193841059654, 343551278125015496821836743879894615130, 1528865864863794364591085900675437568957, 6806154011600221320536026838610543775690, 30309870385160471217429438828604783684063, 135024276473975817820521684494604592141731, 601701174979141349336468667392608849897339, 2682174403773890606526265372622723982425190, 11959869106332140338830195052690800531097055, 53345202397777899584982161676736070962276571, 238007289839348878274854769395491187326642045, 1062203362287527097852702054472301224344629115, 4741811800578950424928585099118659029891054168, 21173712985435421797869641417274671492116243469, 94572040756785404405753140294759200258236622915, 422511574056068242599293004270838091109353156044, 1888085959354717786283706133585602658521134229801, 8439360056193242972860536164684263830692109696182, 37731082848600636076797550747993808179391898396504, 168728527794589612641972306691887871669002871356857, 754700861478336833643175937910891726017883409699806, 3376415828937272767198109519113498411228171724013996, 15108784202343754353082553043313219418247134765894517, 67622847116543212149672016653809818363181959049491016, 302723152820541548810764981884693913650447975187933463, 1355451733660758496354967947418868951913469911129891446, 6070252549522915476104583320335175706103042637591781403, 27190166115146933029643008661744712564550420335584587118, 121814093435277114386490066537472193639982522717702728379, 545835813328045206207128359373935181266277901909199550888, 2446265692149110330731440013351740295531517782222556151059, 10965304063499314074785894561505620014031697789482149965326, 49159968080950882779268712512476824812906808269430381201581, 220432088939645753977407793862627385966338592654222095771064, 988573242539424874740803523335192285385318309940709817336601, 4434169023871365824968387795758266289861020141441754639032938, 19892235135671176171629453571290064549680785495918068012326643, 89252729357086789586217802409624723421641440501639828749128564, 400520438110320470024861816646390077816121929034554917078476105
Implicit equation for the generating function in Maple syntax:
x^2*(x-1)^2*F(x)^3-x*(3*x^2-4*x+2)*F(x)^2+(x+1)*(x-1)^2*F(x)-(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x -1\right)^{2} F \! \left(x \right)^{3}-x \left(3 x^{2}-4 x +2\right) F \! \left(x \right)^{2}+\left(x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(6) = 279
a(7) = 1043
a(8) = 3979
a(9) = 15464
a(10) = 61035
a(11) = 243956
a(12) = 985332
a(n+13) = -6/25*(n+2)*(2*n+1)/(n+14)/(n+13)*a(n)+2/25*(44*n^2+110*n+69)/(n+14)/(n+13)*a(n+1)+2/25*(160*n^2+1926*n+3869)/(n+14)/(n+13)*a(n+2)-1/25*(5273*n^2+50735*n+114018)/(n+14)/(n+13)*a(n+3)+2/25*(11181*n^2+120515*n+317741)/(n+14)/(n+13)*a(n+4)-1/25*(51779*n^2+639265*n+1956368)/(n+14)/(n+13)*a(n+5)+2/25*(38713*n^2+545045*n+1910803)/(n+14)/(n+13)*a(n+6)-2/25*(40080*n^2+637090*n+2525749)/(n+14)/(n+13)*a(n+7)+2/25*(29561*n^2+524608*n+2323625)/(n+14)/(n+13)*a(n+8)-4/25*(7783*n^2+152573*n+746767)/(n+14)/(n+13)*a(n+9)+2/25*(5727*n^2+122867*n+658286)/(n+14)/(n+13)*a(n+10)-1/25*(2791*n^2+65025*n+378356)/(n+14)/(n+13)*a(n+11)+4/5*(20*n+243)/(n+14)*a(n+12), n >= 13
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) = 21
a \! \left(5\right) = 76
a \! \left(6\right) = 279
a \! \left(7\right) = 1043
a \! \left(8\right) = 3979
a \! \left(9\right) = 15464
a \! \left(10\right) = 61035
a \! \left(11\right) = 243956
a \! \left(12\right) = 985332
a \! \left(n +13\right) = -\frac{6 \left(n +2\right) \left(2 n +1\right) a \! \left(n \right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(44 n^{2}+110 n +69\right) a \! \left(n +1\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(160 n^{2}+1926 n +3869\right) a \! \left(n +2\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(5273 n^{2}+50735 n +114018\right) a \! \left(n +3\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(11181 n^{2}+120515 n +317741\right) a \! \left(n +4\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(51779 n^{2}+639265 n +1956368\right) a \! \left(n +5\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(38713 n^{2}+545045 n +1910803\right) a \! \left(n +6\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{2 \left(40080 n^{2}+637090 n +2525749\right) a \! \left(n +7\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(29561 n^{2}+524608 n +2323625\right) a \! \left(n +8\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{4 \left(7783 n^{2}+152573 n +746767\right) a \! \left(n +9\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(5727 n^{2}+122867 n +658286\right) a \! \left(n +10\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(2791 n^{2}+65025 n +378356\right) a \! \left(n +11\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{4 \left(20 n +243\right) a \! \left(n +12\right)}{5 \left(n +14\right)}, \quad n \geq 13
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/277/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[8,x]+F[9,x]
F[8,x] = F[0,x]*F[4,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[15,x]*F[18,x]
F[11,x] = F[12,x]
F[12,x] = F[0,x]*F[13,x]*F[14,x]
F[13,x] = F[0,x]+F[11,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[18,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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)+F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right) F_{4}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{0}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{0}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \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_14(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_14(x)*F_7(x))
Eq(F_7(x), F_8(x) + F_9(x))
Eq(F_8(x), F_0(x)*F_4(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_15(x)*F_18(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_0(x)*F_13(x)*F_14(x))
Eq(F_13(x), F_0(x) + F_11(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_18(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 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, 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, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [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]]}], 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