02341_02413_02431_03241_20341_20413_20431
Counting sequence:
1, 1, 2, 6, 24, 113, 581, 3149, 17688, 102001, 600303, 3590921, 21768532, 133438243, 825696844, 5150735442, 32356470593, 204511706326, 1299656701871, 8299131182069, 53224431071103, 342671103498219, 2213972068163413, 14350163015085861, 93284901748124244, 608037623967390570, 3973026285497195222, 26019699731786911848, 170765455904355035040, 1122921736758291709193, 7397623439283332400981, 48817505031561978445909, 322665105173903932969897, 2135887434530690001822333, 14158441177626691512086291, 93978275090536216871715647, 624570877364819762382003846, 4155738591421682505671128894, 27682121234166383316172974775, 184590674644577123857192558221, 1232128301247363390522949370689, 8232205204373308791653617036225, 55051657091615808943755760430293, 368468076371578817141527931638093, 2468238991703214075625831845152531, 16546883572838235920493905147550215, 111012463805699756886816507992951441, 745314637907380593740292023985167725, 5007335496097423166238433100004882553, 33663590669102328508376759751871795161, 226458813004000215233814360306861745578, 1524340860213132968310989445121658687414, 10266648826535458276251573155661304464553, 69186178764083058272379252107165511691790, 466492668852950409396312007025163593576153, 3146998022390970263979340306269477096004739, 21240565998737195107267483763628211088234856, 143431951184993459557910585376643020364630823, 969010713391714684315446114512324661488658571, 6549484617881956681480664553837664754643641745, 44286863974594208613787545634040001617976708600, 299588882154067762800504138250045972998491557971, 2027465329734879720574181956360866106814201837052, 13726267739226722922272199494602902080877215563202, 92964546963175808882665527900357508999241409067279, 629858421739913239897432392319595927251715145652905, 4268981611062931897890652346269161291068766707450307, 28943875353416171675063115565875052616417848774526933, 196306928601081673773369989464365410076886753512504358, 1331854648657470105696311423835274780845384410458819499, 9038912325579591308216300031318066938060726503028480112, 61363443066354958823016600286893651243908015079674148754, 416709783087102823357769033424546290841478474057502949942, 2830639024350564728898945792994627218598073452265857846623, 19233513772747791249327457047554513008902571001372103195959, 130723238784744308979695787930774750443328743164115792050271, 888717630552636199250466952565921131301846471923182199895026, 6043500391703142059766994372437053014422474734334685801735046, 41107789789574928731976093563583321634295136481676793780389639, 279684035854616469283062344217061769774596987341145347959150981, 1903340458419225694847567659709764838442526799863226752549627719, 12955909493216289482798715334766989536866294902059152611169830309, 88210326122765450136016415725172117354896349637939848565613810629, 600715184104112448112373387192238541617394619093191329322115842097, 4091787796058501357536081056385716481827395373870347069742345705091, 27877295727430058265392234559936121825457994621451380969933699968893, 189967384371108325881504613204777851896815724453716422387538466901463, 1294780637900269649030481448930399092777816307872713397258789736427211, 8826734544296957980055668656754261561229047277932178065839306612977843, 60185058123529491188038630477323244229120508862668596763132628412627929, 410449896955257242270439461980029128340543282474774430646422911721749897, 2799707290638910253795274196775999713758735978129859693404615217634244073, 19100482533309832610539799454634802965949213453934825726599407003503346951, 130332751447819548230607648422293164996353365439011832220607233337061795911, 889485020549951540377328673171654101331772230326997423158509797934441136089, 6071527311532024095590297422500127090759179713371658136230780324925524490693, 41450516478581027987873161076523057368455294011876450497831037646868818878884, 283030414407011307461664908389599915388404137985572515608345320875648391512783, 1932884800943424879561877923427948761172573275953373084024707828775696105293506, 13202224624241163745693409953590838393620785003751211948619770700253943194592604, 90189331918881175829925895824296698086806206067377064077818830903281062674596079
Implicit equation for the generating function in Maple syntax:
(x-1)*F(x)^4+(-x+2)*F(x)^3+(-x-1)*F(x)^2+F(x)-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) F \! \left(x \right)^{4}+\left(-x +2\right) F \! \left(x \right)^{3}+\left(-x -1\right) F \! \left(x \right)^{2}+F \! \left(x \right)-1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 113
a(6) = 581
a(7) = 3149
a(8) = 17688
a(9) = 102001
a(10) = 600303
a(11) = 3590921
a(n+12) = 1600/8897*n*(n+1)*(2*n+1)/(n+10)/(n+12)/(n+11)*a(n)+40/8897*(n+1)*(1814*n^2+4123*n+2403)/(n+10)/(n+12)/(n+11)*a(n+1)-30/8897*(2178*n^3-12697*n^2-81881*n-96050)/(n+10)/(n+12)/(n+11)*a(n+2)+2/8897*(376384*n^3+3722439*n^2+12425549*n+13912914)/(n+10)/(n+12)/(n+11)*a(n+3)-12/8897*(532131*n^3+7201629*n^2+32240951*n+47807794)/(n+10)/(n+12)/(n+11)*a(n+4)+4/8897*(4749038*n^3+75272808*n^2+397041913*n+697149048)/(n+10)/(n+12)/(n+11)*a(n+5)-6/8897*(4978800*n^3+90328153*n^2+546634963*n+1103500758)/(n+10)/(n+12)/(n+11)*a(n+6)+1/8897*(28463783*n^3+583249173*n^2+3987133012*n+9093976596)/(n+10)/(n+12)/(n+11)*a(n+7)-1/8897*(17194963*n^3+393821559*n^2+3007073090*n+7655825904)/(n+10)/(n+12)/(n+11)*a(n+8)+2/8897*(3278329*n^3+83200071*n^2+703342100*n+1980771132)/(n+10)/(n+12)/(n+11)*a(n+9)-2/8897*(750095*n^2+13435303*n+60158832)/(n+12)/(n+11)*a(n+10)+1/8897*(182641*n+1732056)/(n+12)*a(n+11), n >= 12
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) = 24
a \! \left(5\right) = 113
a \! \left(6\right) = 581
a \! \left(7\right) = 3149
a \! \left(8\right) = 17688
a \! \left(9\right) = 102001
a \! \left(10\right) = 600303
a \! \left(11\right) = 3590921
a \! \left(n +12\right) = \frac{1600 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{40 \left(n +1\right) \left(1814 n^{2}+4123 n +2403\right) a \! \left(n +1\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{30 \left(2178 n^{3}-12697 n^{2}-81881 n -96050\right) a \! \left(n +2\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(376384 n^{3}+3722439 n^{2}+12425549 n +13912914\right) a \! \left(n +3\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{12 \left(532131 n^{3}+7201629 n^{2}+32240951 n +47807794\right) a \! \left(n +4\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(4749038 n^{3}+75272808 n^{2}+397041913 n +697149048\right) a \! \left(n +5\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(4978800 n^{3}+90328153 n^{2}+546634963 n +1103500758\right) a \! \left(n +6\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(28463783 n^{3}+583249173 n^{2}+3987133012 n +9093976596\right) a \! \left(n +7\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{\left(17194963 n^{3}+393821559 n^{2}+3007073090 n +7655825904\right) a \! \left(n +8\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(3278329 n^{3}+83200071 n^{2}+703342100 n +1980771132\right) a \! \left(n +9\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{2 \left(750095 n^{2}+13435303 n +60158832\right) a \! \left(n +10\right)}{8897 \left(n +12\right) \left(n +11\right)}+\frac{\left(182641 n +1732056\right) a \! \left(n +11\right)}{8897 n +106764}, \quad n \geq 12
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24141/
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[11,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[11,x]*F[7,x]
F[7,x] = F[12,x]+F[8,x]
F[8,x] = F[0,x]*F[9,x]
F[9,x] = 1/F[11,x]*F[10,x]
F[10,x] = F[2,x]
F[11,x] = x
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]*F[17,x]*F[4,x]
F[14,x] = F[15,x]+F[9,x]
F[15,x] = F[16,x]
F[16,x] = F[9,x]^2*F[11,x]
F[17,x] = F[18,x]+F[9,x]
F[18,x] = F[19,x]
F[19,x] = F[11,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_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{11}\! \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_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = \frac{F_{10}\! \left(x \right)}{F_{11}\! \left(x \right)}
F_{10}\! \left(x \right) = F_{2}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \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_3(x))
Eq(F_3(x), F_11(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_11(x)*F_7(x))
Eq(F_7(x), F_12(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x))
Eq(F_9(x), F_10(x)/F_11(x))
Eq(F_10(x), F_2(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)*F_17(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_9(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_11(x)*F_9(x)**2)
Eq(F_17(x), F_18(x) + F_9(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_14(x)*F_17(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"class_module": "comb_spec_searcher.strategies.rule", "idx": 1, "original_rule": {"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, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0], [0, 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