01342_01432_10342_10432_20341_20431
Counting sequence:
1, 1, 2, 6, 24, 114, 596, 3294, 18852, 110486, 658808, 3980626, 24305912, 149698890, 928690892, 5797176310, 36383341132, 229429650654, 1452882085616, 9235440975050, 58908000805664, 376918048829266, 2418583073552420, 15560267944268302, 100352465017605620, 648659932662446374, 4201610976971133480, 27268625154420769666, 177298201073964510408, 1154753894376116109402, 7533095585553577557564, 49217026538146348080742, 322016069734878542931484, 2109723437116604817851246, 13839729648108635351567136, 90898042177795252535981946, 597694484972741083103971312, 3934383614201061392567208546, 25925289999022945220418386708, 171000888422425801877910469374, 1128965753702816539977972347076, 7460226420357404588706873148854, 49339472323649150380946388146136, 326581653191746184661375188325042, 2163358184350281392927995192790680, 14341347851206997340295024848044906, 95139924481504684911371346928872620, 631588393389414272756857524738146774, 4195579394163355547627393701032342508, 27888481272718729097915077961310362110, 185490724010315225450163307786100262480, 1234450923922888927182880724886371538730, 8219969493838205046027112100396438594240, 54764898216873426379421409244027555760498, 365057692615066200962518964922373762652420, 2434668014837300918542254634180774056137966, 16245359470990372564686212024077271516073364, 108448343896966140639869985760426700080951878, 724291861045998393561611914532349482659343496, 4839436194373500290587218369109180008870178530, 32348946000477973533388118842063792998068891240, 216323512646685059886982461810671249597449544314, 1447170982087371632059667658320973355932790468124, 9685076626755494541729117443094348409474451127878, 64840773943364815526433244253603472369467994796988, 434260455839568247442796486350868566710569431275662, 2909407911031120885282718238014276234479245275170944, 19498745237334753607754039820746562168871159020063706, 130723031139103693820321096747890756813610050407736208, 876671344128923499309579544868840275296564875610995842, 5881076197677848448387698422930587943263370644273603060, 39464653867494270014348978770818043821076029538304921566, 264903468692615870849373553467679754230211216947584070756, 1778653511864764358189019046087273677452571413966464855766, 11945822920387386354499637687992406030259464615411843555384, 80252501913663795597715428381061986898183012969178094503954, 539281825614019499627828215465267439389734011596405250063416, 3624805512326185536174557680744973467292385264494235674144138, 24370390936933380458911618432885976774759034135869749277528716, 163887711488111827681126344968533954607155715578620596118176694, 1102386139918384400050068581922104455447003401113171644607395212, 7416892835934867241682258212052265644392309419499306337257467678, 49912429624453681003030926244009751764041553571121856460418572720, 335963033343011444537152079912160794861327786793113025113857934730, 2261872218135635135026223636544824491531438154472874264170906450528, 15231273704625776797578669700365902178405672341585880315865582465938, 102587355030537315181220143138026602959478349408469112375949682664932, 691096734692543430844301148715477374793729638030173691907224782674894, 4656603508584297195191071102283724153612667876221284877664916255388980, 31382184644056684838606784556976604135062155258567471833668512711142246, 211533289119295779930038609707383246571622025142364390178767026966676200, 1426113572839919420584490565100629994453058760293824755936925178011471426, 9616291287268042663015075294439984836734508017214564339825974098345207816, 64854114285338371164890863970605012279164030419766173586042110972693034138, 437463933759540580353954782515866905288684385610838454992481321902549445116, 2951346488654363690702515247199777461761755235531988433938097000235814879782, 19914519899123146148478091633213816996816023184813660652653123976822531082588, 134397039184533821759767731408521443751179957292509106595753634538755650309550, 907148414361440781652612420491468302512891661829624435223604009962153280971232, 6123988751536980768931684615633697528141816301319936907661069676182109166842938, 41348186645847531779187710226095684568216660344913846980834084958512359051089712
Implicit equation for the generating function in Maple syntax:
(2*x^3-x^2-x+1)*F(x)^4+(-x^3-2*x^2+6*x-5)*F(x)^3+(4*x^2-9*x+9)*F(x)^2+(-x^2+4*x-7)*F(x)+2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-x^{2}-x +1\right) F \! \left(x \right)^{4}+\left(-x^{3}-2 x^{2}+6 x -5\right) F \! \left(x \right)^{3}+\left(4 x^{2}-9 x +9\right) F \! \left(x \right)^{2}+\left(-x^{2}+4 x -7\right) F \! \left(x \right)+2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 114
a(6) = 596
a(7) = 3294
a(8) = 18852
a(9) = 110486
a(10) = 658808
a(11) = 3980626
a(12) = 24305912
a(13) = 149698890
a(14) = 928690892
a(15) = 5797176310
a(16) = 36383341132
a(17) = 229429650654
a(n+18) = -4/3*n*(n+1)*(2*n+1)/(2*n+33)/(n+18)/(n+16)*a(n)+8*(n+1)*(17*n^2+64*n+57)/(2*n+33)/(n+18)/(n+16)*a(n+1)-1/3*(7922*n^3+61755*n^2+156199*n+128448)/(2*n+33)/(n+18)/(n+16)*a(n+2)+(26436*n^3+284669*n^2+1005886*n+1167452)/(2*n+33)/(n+18)/(n+16)*a(n+3)-1/3*(451702*n^3+6211290*n^2+28157234*n+42123609)/(2*n+33)/(n+18)/(n+16)*a(n+4)+(493794*n^3+8284088*n^2+45873865*n+84001439)/(2*n+33)/(n+18)/(n+16)*a(n+5)-(855090*n^3+17002472*n^2+111165709*n+239926235)/(2*n+33)/(n+18)/(n+16)*a(n+6)+1/3*(1353200*n^3+32286693*n^2+243072904*n+587342763)/(2*n+33)/(n+18)/(n+16)*a(n+7)+(741902*n^3+17085385*n^2+138338296*n+390297247)/(2*n+33)/(n+18)/(n+16)*a(n+8)-(892386*n^3+21433717*n^2+175407309*n+488519437)/(2*n+33)/(n+18)/(n+16)*a(n+9)+1/3*(271670*n^3+3484188*n^2-1521935*n-99617496)/(2*n+33)/(n+18)/(n+16)*a(n+10)+(271850*n^3+8121669*n^2+80637130*n+265838784)/(2*n+33)/(n+18)/(n+16)*a(n+11)-1/3*(253120*n^3+7442178*n^2+71530322*n+223027245)/(2*n+33)/(n+18)/(n+16)*a(n+12)-1/3*(24148*n^3+1068744*n^2+15153587*n+69645555)/(2*n+33)/(n+18)/(n+16)*a(n+13)+1/3*(14240*n^3+538086*n^2+6710851*n+27579705)/(2*n+33)/(n+18)/(n+16)*a(n+14)+(362*n^3+19015*n^2+323488*n+1795483)/(2*n+33)/(n+18)/(n+16)*a(n+15)-(316*n^3+14705*n^2+228064*n+1178845)/(2*n+33)/(n+18)/(n+16)*a(n+16)+(44*n^3+2125*n^2+34189*n+183255)/(2*n+33)/(n+18)/(n+16)*a(n+17), n >= 18
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) = 114
a \! \left(6\right) = 596
a \! \left(7\right) = 3294
a \! \left(8\right) = 18852
a \! \left(9\right) = 110486
a \! \left(10\right) = 658808
a \! \left(11\right) = 3980626
a \! \left(12\right) = 24305912
a \! \left(13\right) = 149698890
a \! \left(14\right) = 928690892
a \! \left(15\right) = 5797176310
a \! \left(16\right) = 36383341132
a \! \left(17\right) = 229429650654
a \! \left(n +18\right) = -\frac{4 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{8 \left(n +1\right) \left(17 n^{2}+64 n +57\right) a \! \left(n +1\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(7922 n^{3}+61755 n^{2}+156199 n +128448\right) a \! \left(n +2\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(26436 n^{3}+284669 n^{2}+1005886 n +1167452\right) a \! \left(n +3\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(451702 n^{3}+6211290 n^{2}+28157234 n +42123609\right) a \! \left(n +4\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(493794 n^{3}+8284088 n^{2}+45873865 n +84001439\right) a \! \left(n +5\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(855090 n^{3}+17002472 n^{2}+111165709 n +239926235\right) a \! \left(n +6\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(1353200 n^{3}+32286693 n^{2}+243072904 n +587342763\right) a \! \left(n +7\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(741902 n^{3}+17085385 n^{2}+138338296 n +390297247\right) a \! \left(n +8\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(892386 n^{3}+21433717 n^{2}+175407309 n +488519437\right) a \! \left(n +9\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(271670 n^{3}+3484188 n^{2}-1521935 n -99617496\right) a \! \left(n +10\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(271850 n^{3}+8121669 n^{2}+80637130 n +265838784\right) a \! \left(n +11\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(253120 n^{3}+7442178 n^{2}+71530322 n +223027245\right) a \! \left(n +12\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(24148 n^{3}+1068744 n^{2}+15153587 n +69645555\right) a \! \left(n +13\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(14240 n^{3}+538086 n^{2}+6710851 n +27579705\right) a \! \left(n +14\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(362 n^{3}+19015 n^{2}+323488 n +1795483\right) a \! \left(n +15\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(316 n^{3}+14705 n^{2}+228064 n +1178845\right) a \! \left(n +16\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(44 n^{3}+2125 n^{2}+34189 n +183255\right) a \! \left(n +17\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}, \quad n \geq 18
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24099/
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[17,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[18,x]+F[8,x]
F[8,x] = F[0,x]*F[9,x]
F[9,x] = F[0,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[17,x]
F[12,x] = 1/F[17,x]*F[13,x]
F[13,x] = F[14,x]
F[14,x] = -F[0,x]+F[15,x]
F[15,x] = 1/F[17,x]*F[16,x]
F[16,x] = F[2,x]
F[17,x] = x
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[20,x]*F[4,x]
F[20,x] = 1/F[17,x]*F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[17,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[17,x]*F[23,x]
F[27,x] = F[28,x]
F[28,x] = F[17,x]*F[23,x]*F[29,x]
F[29,x] = F[24,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[17,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[24,x]*F[29,x]
F[34,x] = F[35,x]
F[35,x] = F[17,x]*F[29,x]*F[32,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_{17}\! \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_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{18}\! \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) = F_{0}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{12}\! \left(x \right) = \frac{F_{13}\! \left(x \right)}{F_{17}\! \left(x \right)}
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = -F_{0}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = \frac{F_{16}\! \left(x \right)}{F_{17}\! \left(x \right)}
F_{16}\! \left(x \right) = F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = \frac{F_{21}\! \left(x \right)}{F_{17}\! \left(x \right)}
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right) F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{29}\! \left(x \right) F_{32}\! \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_17(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_17(x)*F_7(x))
Eq(F_7(x), F_18(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x))
Eq(F_9(x), F_0(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_17(x))
Eq(F_12(x), F_13(x)/F_17(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), -F_0(x) + F_15(x))
Eq(F_15(x), F_16(x)/F_17(x))
Eq(F_16(x), F_2(x))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_20(x)*F_4(x))
Eq(F_20(x), F_21(x)/F_17(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_17(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_17(x)*F_23(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_17(x)*F_23(x)*F_29(x))
Eq(F_29(x), F_24(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_17(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_24(x)*F_29(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_17(x)*F_29(x)*F_32(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:
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