02341_02431_03241_20341_20431
Counting sequence:
1, 1, 2, 6, 24, 115, 615, 3526, 21189, 131695, 839493, 5457909, 36049457, 241212167, 1631550587, 11137519970, 76630368626, 530868836388, 3699820449265, 25922597203100, 182485404261527, 1290079098341387, 9155068771487576, 65194072018733706, 465715858839593444, 3336437351090791991, 23965753373255556125, 172565796366150370129, 1245356218639260583868, 9006089509698503021595, 65255759861228849582270, 473678912369449236873885, 3444135417616308120113640, 25081913637695545268210059, 182929206366993693671790124, 1336007305546825720899893785, 9770186529339026481294956409, 71537365828286267887278299234, 524408336218392179560835040169, 3848448174468221131119517860279, 28271971665258230607998600777213, 207901606295406142631246728295221, 1530275060349647888928036598322658, 11273823488374171003639351121025836, 83127478939419217396100256591689506, 613440557526180428127452323027849369, 4530424218836541759758651870476074548, 33483344524523050671664064118319363837, 247644339388747347027089750558431448163, 1832839825261807778600542025505977314498, 13573907553802482432662868240773546024607, 100590740850208975697164062304590527139444, 745886999515443379892560835142286799180114, 5534007950285856784978617339158718803418347, 41081719562777847874155888944561704382584526, 305133904954319978009840288427305718084638688, 2267549437435820292467783159127179783351088988, 16859294476818333712207045903626988371545599422, 125409571930230555515069990415361521648840220070, 933304490632341180684907345151883330033717523531, 6948811082721843477511208162839079801989459061576, 51758964624916572635597376826682863470622494454529, 385693531172101938626115207657087717142390069841419, 2875245327581570538241975428169066479485105900403295, 21442607540395398898910778412235956362364417432297541, 159972398324870109431336297415280785202802259262199335, 1193911576215279533408032142218538250092195533585655319, 8913618318784647540242923864276448140183647557959077895, 66571141379572878577214087714658642393837059079828335667, 497351709500917344610244027813442147121719370595816306665, 3716914820322015775487129044310488407915872345060814934732, 27786823801681206723392085177823713208248535071768117108520, 207791897106048706915006442686941077718008325762271725304305, 1554346976750008680230876298747544694078069463883312787546318, 11630366955909858147794331796866671131240055181197327451387394, 87048542816190057909283651989098410982499043756546726535730686, 651701826298358993789824863003437508813762453330918579727443671, 4880367746373555185534328017171051672726916832887418283254121416, 36556894342619124676011588929108290054528512331243765277632474995, 273902659561747110030282601880243392658962071259684336121429570466, 2052724201588034078647977956562045912154929821406497371163994090749, 15387556038914395106943584010737600607653724942069507120944794484847, 115374751502092886500335597270875276207009881797504364008591786982446, 865269681654457342054526842540299041878260171877157190717628808889136, 6490667378760685817394361610904527456015476449091364557907252075368630, 48699224933220618310662326005540106940474097576586571858738942858760447, 365466290574214022575110534149655149740561900553163377482670027539858113, 2743234996868962508260977414346463442189454894440731603755673850828928802, 20595247626250846505366161002776119185572688795011728462099684939474131553, 154652633477159965572072185949894105386134991668900389042292451495029316125, 1161534101085723936722388866153602135527314385774396518554645801199203706255, 8725474734660387927883573333117594667259908106452733555626857562513453114383, 65558166207631894937172909932944098398829741538362596342554653449197890925011, 492655567081161922494471797421115558296638499136438766329316412379210753616239, 3702858795650606741092074793975511994965760682247716364339831033470350206683029, 27835973422350959944674739639206577601318972725044831172621780532644477856851371, 209290531539485366713358007861936783640973710935120668321096774873549367911617697, 1573856400843858673168274215187744761613028586344712654144994426788081858764571463, 11837268064286095151270569830964463730639846698347074183947448441603077119566744452, 89044532817654563248880941541645092914284616203160233918044663495143258821622463179, 669932465818189221702800021566409394969275226070570389930990129103429138307760215404
Implicit equation for the generating function in Maple syntax:
(x^3-x+1)*F(x)^4+(-x^2+3*x-4)*F(x)^3+(x^2-3*x+6)*F(x)^2+(x-4)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-x +1\right) F \! \left(x \right)^{4}+\left(-x^{2}+3 x -4\right) F \! \left(x \right)^{3}+\left(x^{2}-3 x +6\right) F \! \left(x \right)^{2}+\left(x -4\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) = 24
a(5) = 115
a(6) = 615
a(7) = 3526
a(8) = 21189
a(9) = 131695
a(10) = 839493
a(11) = 5457909
a(12) = 36049457
a(n+13) = 21*(2*n+1)*(n+2)*(n+1)/(3*n+35)/(3*n+37)/(n+12)*a(n)-1/3*(n+2)*(2032*n^2+9181*n+9105)/(3*n+35)/(3*n+37)/(n+12)*a(n+1)+2/9*(21533*n^3+202824*n^2+621154*n+620787)/(3*n+35)/(3*n+37)/(n+12)*a(n+2)-1/2*(36501*n^3+457060*n^2+1866067*n+2496804)/(3*n+35)/(3*n+37)/(n+12)*a(n+3)+1/18*(711623*n^3+11792234*n^2+62925435*n+109452768)/(3*n+35)/(3*n+37)/(n+12)*a(n+4)-1/9*(388612*n^3+9114363*n^2+64449851*n+144303840)/(3*n+35)/(3*n+37)/(n+12)*a(n+5)+1/9*(81116*n^3+6193479*n^2+66902801*n+198124938)/(3*n+35)/(3*n+37)/(n+12)*a(n+6)+1/18*(1957259*n^3+34807718*n^2+207880287*n+418555764)/(3*n+35)/(3*n+37)/(n+12)*a(n+7)-1/18*(2592129*n^3+54685088*n^2+386580871*n+917727144)/(3*n+35)/(3*n+37)/(n+12)*a(n+8)+1/18*(1382477*n^3+30678056*n^2+225553835*n+549383232)/(3*n+35)/(3*n+37)/(n+12)*a(n+9)+1/6*(4331*n^3+352140*n^2+5256837*n+21911508)/(3*n+35)/(3*n+37)/(n+12)*a(n+10)-4/3*(3385*n^3+103002*n^2+1043715*n+3521684)/(3*n+35)/(3*n+37)/(n+12)*a(n+11)+(513*n^3+16632*n^2+179579*n+645716)/(3*n+35)/(3*n+37)/(n+12)*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) = 24
a \! \left(5\right) = 115
a \! \left(6\right) = 615
a \! \left(7\right) = 3526
a \! \left(8\right) = 21189
a \! \left(9\right) = 131695
a \! \left(10\right) = 839493
a \! \left(11\right) = 5457909
a \! \left(12\right) = 36049457
a \! \left(n +13\right) = \frac{21 \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{\left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(n +2\right) \left(2032 n^{2}+9181 n +9105\right) a \! \left(n +1\right)}{3 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{2 \left(21533 n^{3}+202824 n^{2}+621154 n +620787\right) a \! \left(n +2\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(36501 n^{3}+457060 n^{2}+1866067 n +2496804\right) a \! \left(n +3\right)}{2 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(711623 n^{3}+11792234 n^{2}+62925435 n +109452768\right) a \! \left(n +4\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(388612 n^{3}+9114363 n^{2}+64449851 n +144303840\right) a \! \left(n +5\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(81116 n^{3}+6193479 n^{2}+66902801 n +198124938\right) a \! \left(n +6\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(1957259 n^{3}+34807718 n^{2}+207880287 n +418555764\right) a \! \left(n +7\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(2592129 n^{3}+54685088 n^{2}+386580871 n +917727144\right) a \! \left(n +8\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(1382477 n^{3}+30678056 n^{2}+225553835 n +549383232\right) a \! \left(n +9\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(4331 n^{3}+352140 n^{2}+5256837 n +21911508\right) a \! \left(n +10\right)}{6 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{4 \left(3385 n^{3}+103002 n^{2}+1043715 n +3521684\right) a \! \left(n +11\right)}{3 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(513 n^{3}+16632 n^{2}+179579 n +645716\right) a \! \left(n +12\right)}{\left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}, \quad n \geq 13
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/24082/
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[23,x]*F[4,x]
F[4,x] = F[5,x,1]
F[5,x,k[0]] = F[0,x]+F[6,x,k[0]]
F[6,x,k[0]] = F[14,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]
F[8,x,k[0]] = F[13,x,k[0]]*F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[11,x,k[0]]
F[10,x,k[0]] = F[1,x]+F[7,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[10,x,k[0]]^2*F[13,x,k[0]]*F[9,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x,k[0]] = F[15,x,k[0]]
F[15,x,k[0]] = F[16,x,k[0]]*F[23,x]
F[16,x,k[0]] = F[17,x,k[0]]+F[18,x,k[0]]
F[17,x,k[0]] = F[4,x]*F[7,x,k[0]]
F[18,x,k[0]] = F[19,x,k[0]]
F[19,x,k[0]] = F[13,x,k[0]]*F[20,x,k[0]]*F[9,x,k[0]]
F[21,x,k[0]] = F[13,x,k[0]]*F[20,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = -k[0]*(F[6,x,1]-F[6,x,k[0]])/(-1+k[0])
F[23,x] = 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_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x , 1\right)
F_{5}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)
F_{6}\! \left(x , y\right) = F_{14}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{10}\! \left(x , y\right)^{2} F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{23}\! \left(x \right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}
F_{23}\! \left(x \right) = x
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_23(x)*F_4(x))
Eq(F_4(x), F_5(x, 1))
Eq(F_5(x, k_0), F_0(x) + F_6(x, k_0))
Eq(F_6(x, k_0), F_14(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0))
Eq(F_8(x, k_0), F_13(x, k_0)*F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_11(x, k_0))
Eq(F_10(x, k_0), F_1(x) + F_7(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_10(x, k_0)**2*F_13(x, k_0)*F_9(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x, k_0), F_15(x, k_0))
Eq(F_15(x, k_0), F_16(x, k_0)*F_23(x))
Eq(F_16(x, k_0), F_17(x, k_0) + F_18(x, k_0))
Eq(F_17(x, k_0), F_4(x)*F_7(x, k_0))
Eq(F_18(x, k_0), F_19(x, k_0))
Eq(F_19(x, k_0), F_13(x, k_0)*F_20(x, k_0)*F_9(x, k_0))
Eq(F_21(x, k_0), F_13(x, k_0)*F_20(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), -F_6(x, 1) + (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_23(x), 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, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"class_module": "comb_spec_searcher.strategies.rule", "idx": 0, "original_rule": {"children": [{"assumptions": [{"assumption": "TrackingAssumption", "class_module": "tilings.assumptions", "gps": [{"patt": [0], "pos": [[2, 0]]}, {"patt": [0], "pos": [[2, 1]]}]}], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 1], [1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], 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