0132_0231_1023
Counting sequence:
1, 1, 2, 6, 21, 76, 278, 1026, 3818, 14308, 53932, 204273, 776859, 2964716, 11348261, 43552751, 167535302, 645786439, 2493841305, 9646413500, 37369009359, 144959573495, 563015802889, 2189212260530, 8521370179069, 33200950530200, 129473317107039, 505325461532842, 1973779259965682, 7715077590253368, 30177017325511710, 118110494501739609, 462551765394050587, 1812495148105932089, 7105984930850569065, 27873406043103496254, 109386206384479049211, 429468369683086984912, 1686887308915733901723, 6628543643968270472912, 26056668791300485377428, 102466181052367357020868, 403084222163241947337872, 1586199614256831336932789, 6243963053777816978576036, 24586534351256875873813470, 96841889349499443090296241, 381551286213594104878819326, 1503700768187701030375201431, 5927671738410420263906788061, 23373123524259766370041250904, 92183916897012881144028782440, 363659965241641209595268910885, 1434941220284644726076185715181, 5663273818433510588466192865469, 22355921174775685514192197392293, 88268559669629973145753209979757, 348582058217318985240522410041030, 1376850369870098523070069660891328, 5439369117895023073689238698238934, 21492545745600896191905143753298295, 84938057531558988534939116335325819, 335729599755840890441593565981553568, 1327234297627571597863790766074192833, 5247761950861229279116909989529923486, 20752347406053067871580768796386700006, 82077665884374448661097637619000670820, 324672576482035081208062370659912248683, 1284479900295315686027803244414610867516, 5082395303515807257899921148258623089158, 20112561685927926521525694174968344585611, 79601750739619395601164779304591149706068, 315088575722928947739466823338324963755539, 1247372269335042631147624732254804617964487, 4938687446925562001419176358369833683669158, 19555896044327565818153018120182499418059178, 77444995792860165913946648471171029581141233, 306730704931987095711339711250731371512043011, 1214977487136028546614450020268085175706522853, 4813103177834081127262609082242917367115072811, 19068959362793724531541675678590010841259986631, 75556642669093099561629364986285986800587812492, 299406442690776371335294835545855219923618990460, 1186564880317143558981745671463695863874408044398, 4702867687110378317376090883682118037852567127873, 18641207735262738966673305544860581115879332725081, 73896602685894583020638166202260029684418767982490, 292963282837591155557467845176173438990668804769355, 1161553962402141841349923376304329558304348561415025, 4605770274237784855633830652183123128596962338466219, 18264217614567669066971912340039839200477101943081460, 72432762774182008084250411951553852317371997780422365, 287278739803308125117234549701760270415818741974135076, 1139477322163973638879073990960169405769058170869121495, 4520026427376262060725613584929704934350174913717119274, 17931172822263377082079305579557854769490357830872970978, 71139075856523686950930934092246860352816018986076124394, 282253234293279674576089552935401417320704653412646336996, 1119954098432667962569937941619152492340042824470828123637, 4444178850739824724554515271058973407244728845049060086420, 17636494979953251278976388811586993073693972969989495732626
Generating function in Maple syntax:
(-(x^2-x+1)*(x-1)^2*(1-4*x)^(1/2)+x^4+x^3-4*x^2+5*x-1)/(2*x^5-8*x^4+18*x^3-20*x^2+12*x-2)
Generating function in latex syntax:
\frac{-\left(x^{2}-x +1\right) \left(x -1\right)^{2} \sqrt{1-4 x}+x^{4}+x^{3}-4 x^{2}+5 x -1}{2 x^{5}-8 x^{4}+18 x^{3}-20 x^{2}+12 x -2}
Generating function in sympy syntax:
(x**4 + x**3 - 4*x**2 + 5*x + sqrt(1 - 4*x)*(x - 1)**2*(-x**2 + x - 1) - 1)/(2*x**5 - 8*x**4 + 18*x**3 - 20*x**2 + 12*x - 2)
Implicit equation for the generating function in Maple syntax:
(x^5-4*x^4+9*x^3-10*x^2+6*x-1)*F(x)^2+(-x^4-x^3+4*x^2-5*x+1)*F(x)+x^2*(x^2-2*x+2) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-4 x^{4}+9 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)^{2}+\left(-x^{4}-x^{3}+4 x^{2}-5 x +1\right) F \! \left(x \right)+x^{2} \left(x^{2}-2 x +2\right) = 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) = 278
a(7) = 1026
a(8) = 3818
a(n+9) = 2*(1+2*n)/(n+9)*a(n)-(34+25*n)/(n+9)*a(n+1)+(171+82*n)/(n+9)*a(n+2)-(493+167*n)/(n+9)*a(n+3)+(894+229*n)/(n+9)*a(n+4)-9*(119+24*n)/(n+9)*a(n+5)+3*(277+46*n)/(n+9)*a(n+6)-(397+56*n)/(n+9)*a(n+7)+(97+12*n)/(n+9)*a(n+8), n >= 9
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) = 278
a \! \left(7\right) = 1026
a \! \left(8\right) = 3818
a \! \left(n +9\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +9}-\frac{\left(34+25 n \right) a \! \left(n +1\right)}{n +9}+\frac{\left(171+82 n \right) a \! \left(n +2\right)}{n +9}-\frac{\left(493+167 n \right) a \! \left(n +3\right)}{n +9}+\frac{\left(894+229 n \right) a \! \left(n +4\right)}{n +9}-\frac{9 \left(119+24 n \right) a \! \left(n +5\right)}{n +9}+\frac{3 \left(277+46 n \right) a \! \left(n +6\right)}{n +9}-\frac{\left(397+56 n \right) a \! \left(n +7\right)}{n +9}+\frac{\left(97+12 n \right) a \! \left(n +8\right)}{n +9}, \quad n \geq 9
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/263/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[11,x]*F[3,x]
F[3,x] = F[1,x]+F[22,x]+F[4,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[1,x]+F[12,x]+F[22,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[11,x]*F[3,x]*F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[8,x]
F[11,x] = x
F[12,x] = F[13,x,1]
F[13,x,k[0]] = F[11,x]*F[14,x,k[0]]
F[14,x,k[0]] = F[1,x]+F[15,x,k[0]]+F[17,x,k[0]]+F[20,x,k[0]]+F[22,x]
F[15,x,k[0]] = F[16,x,k[0]]*F[3,x]
F[16,x,k[0]] = k[0]*x
F[17,x,k[0]] = F[18,x,k[0]]
F[18,x,k[0]] = F[11,x]*F[19,x,k[0]]*F[3,x]*F[8,x]
F[19,x,k[0]] = F[1,x]+F[16,x,k[0]]
F[20,x,k[0]] = F[11,x]*F[21,x,k[0]]
F[21,x,k[0]] = (F[14,x,k[0]]*k[0]-F[14,x,1])/(-1+k[0])
F[22,x] = F[11,x]*F[23,x]
F[23,x] = F[24,x,1]
F[24,x,k[0]] = F[1,x]+F[13,x,k[0]]+F[25,x,k[0]]+F[26,x,k[0]]
F[25,x,k[0]] = F[16,x,k[0]]*F[24,x,k[0]]
F[26,x,k[0]] = F[11,x]*F[27,x,k[0]]
F[27,x,k[0]] = (F[24,x,k[0]]*k[0]-F[24,x,1])/(-1+k[0])
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_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x , 1\right)
F_{13}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{22}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{16}\! \left(x , y\right) = y x
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{19}\! \left(x , y\right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x , 1\right)
F_{24}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{25}\! \left(x , y\right)+F_{26}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{24}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}
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_11(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_22(x) + F_4(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_12(x) + F_22(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_11(x)*F_3(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_8(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x, 1))
Eq(F_13(x, k_0), F_11(x)*F_14(x, k_0))
Eq(F_14(x, k_0), F_1(x) + F_15(x, k_0) + F_17(x, k_0) + F_20(x, k_0) + F_22(x))
Eq(F_15(x, k_0), F_16(x, k_0)*F_3(x))
Eq(F_16(x, k_0), k_0*x)
Eq(F_17(x, k_0), F_18(x, k_0))
Eq(F_18(x, k_0), F_11(x)*F_19(x, k_0)*F_3(x)*F_8(x))
Eq(F_19(x, k_0), F_1(x) + F_16(x, k_0))
Eq(F_20(x, k_0), F_11(x)*F_21(x, k_0))
Eq(F_21(x, k_0), (-k_0*F_14(x, k_0) + F_14(x, 1))/(1 - k_0))
Eq(F_22(x), F_11(x)*F_23(x))
Eq(F_23(x), F_24(x, 1))
Eq(F_24(x, k_0), F_1(x) + F_13(x, k_0) + F_25(x, k_0) + F_26(x, k_0))
Eq(F_25(x, k_0), F_16(x, k_0)*F_24(x, k_0))
Eq(F_26(x, k_0), F_11(x)*F_27(x, k_0))
Eq(F_27(x, k_0), (-k_0*F_24(x, k_0) + F_24(x, 1))/(1 - k_0))
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, 3, 2], "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, 0, 2, 3], "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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, 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