0123_0231_0312_1203_2013
Counting sequence:
1, 1, 2, 6, 19, 59, 184, 587, 1921, 6426, 21886, 75667, 264956, 937922, 3351336, 12071815, 43789630, 159821034, 586468261, 2162412817, 8007435590, 29765967861, 111034175619, 415493596046, 1559273972658, 5867131187466, 22130038541992, 83658510392999, 316911106892873, 1202817718311824, 4573404260265315, 17418269553814714, 66443016774665882, 253822603403232097, 970978080862774621, 3719218651821791777, 14263468093206053574, 54764655158138978206, 210499185795091121817, 809935612500466346905, 3119447365037752460865, 12025709783797388838176, 46401285403630134171705, 179191451358525318755647, 692557062183487202477827, 2678730410735461716254282, 10368665094899414150837392, 40162733326235007934853993, 155674440024201789660191223, 603799660486369557307642759, 2343357361402657292189860154, 9100045720843208610501138609, 35358833526847673244508314161, 137464982059797073530799465547, 534708455465407754906805140358, 2080963043765443137508774161340, 8102630025610171914488898914117, 31564170821637181626549143220675, 123016198387020961599679697897851, 479648328069183480481483364824718, 1870982845344957603395201447647883, 7301242216920068761932355835479841, 28503482918534596478694332643409285, 111318604836471488912726105798980297, 434911525272373504604348087008167522, 1699778585992008708565791208911445602, 6645646467334135064861060512161748724, 25991476135390592186136810374613894464, 101687853219823821792080996147158436633, 397967329415667289782094986189448567802, 1557980215332431684765093617446849086286, 6101108814764493107940464063150425700225, 23899247102805507905878244130101226711016, 93645031301325730864284113932439019221252, 367034534108727161794129986292441239384296, 1438956033903648176325712983293508067153592, 5642913497384523610858358637611557271114114, 22134592056411204951156240513531031967809313, 86845839666624618018130528058146465547941846, 340826298200724023703784609269217655412958650, 1337892073903196006068890387977901753106127761, 5253036146897384195187737161977128915510485006, 20629967263014725272302417628180236647638468038, 81036959094724908110457771647781314859230106962, 318391798752665925366459551229576239785328348524, 1251216673624327184169421489503275034459170940114, 4918050002646946886117461465954539938946723947009, 19334860801025358238227252762477812118704922857548, 76028224831274959222729751188177398871680398366662, 299014609087721584241364592787774931182942307988779, 1176228914776231252079392154747636756149364026801066, 4627765824693812983060976758484829144952936738694790, 18210808557778237049344277855185389545426407674128778, 71674343295645541126123787092176992069842113634400822, 282145558513477423053625453128468087702921720451412132, 1110851818084245007376915392622834042512304295614991765, 4374323754175827065471941451778225436925650330064723474, 17228049417785792165726310074928416660915954388708611086, 67862561038894675312894351066756947115160071323593992311, 267357230682637394047982312926087706161301544637960487236, 1053464291187298034795753091153788460928642783428338200844
Generating function in Maple syntax:
-1/2*(-3*x^3+3*x^2+(x-1)^3*(-4*x+1)^(1/2)-3*x+1)/x/(x^6-4*x^5+12*x^4-17*x^3+14*x^2-6*x+1)*(x-1)^3
Generating function in latex syntax:
-\frac{\left(-3 x^{3}+3 x^{2}+\left(x -1\right)^{3} \sqrt{-4 x +1}-3 x +1\right) \left(x -1\right)^{3}}{2 x \left(x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1\right)}
Generating function in sympy syntax:
(x - 1)**3*(3*x**3/2 - 3*x**2/2 + 3*x/2 - sqrt(1 - 4*x)*(x - 1)**3/2 - 1/2)/(x*(x**6 - 4*x**5 + 12*x**4 - 17*x**3 + 14*x**2 - 6*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^6-4*x^5+12*x^4-17*x^3+14*x^2-6*x+1)*F(x)^2-(3*x^3-3*x^2+3*x-1)*(x-1)^3*F(x)+(x-1)^6 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1\right) F \! \left(x \right)^{2}-\left(3 x^{3}-3 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x -1\right)^{6} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 59
a(6) = 184
a(7) = 587
a(n+8) = -2*(1+2*n)/(9+n)*a(n)+(40+21*n)/(9+n)*a(1+n)-23*(7+3*n)/(9+n)*a(n+2)+2*(227+66*n)/(9+n)*a(n+3)-17*(40+9*n)/(9+n)*a(n+4)+(613+111*n)/(9+n)*a(n+5)-16*(20+3*n)/(9+n)*a(n+6)+(86+11*n)/(9+n)*a(n+7), n >= 8
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) = 19
a \! \left(5\right) = 59
a \! \left(6\right) = 184
a \! \left(7\right) = 587
a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(40+21 n \right) a \! \left(1+n \right)}{9+n}-\frac{23 \left(7+3 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(227+66 n \right) a \! \left(n +3\right)}{9+n}-\frac{17 \left(40+9 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(613+111 n \right) a \! \left(n +5\right)}{9+n}-\frac{16 \left(20+3 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(86+11 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8
Specification 1
Strategy pack name: insertion_point_placements_tracked_fusion
Tree: http://permpal.com/tree/21528/
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[4,x]*F[8,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[12,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[10,x]*F[8,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[13,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[8,x]
F[19,x] = F[20,x]+F[51,x]
F[20,x] = F[21,x]+F[48,x]
F[21,x] = F[13,x]+F[22,x]
F[22,x] = F[23,x,1]
F[23,x,k[0]] = F[2,x]+F[24,x,k[0]]
F[24,x,k[0]] = F[25,x,k[0]]
F[25,x,k[0]] = F[26,x,k[0]]*F[32,x,k[0]]
F[26,x,k[0]] = F[23,x,k[0]]+F[27,x,k[0]]
F[27,x,k[0]] = F[28,x,k[0]]+F[33,x,k[0]]
F[28,x,k[0]] = F[13,x]*F[29,x,k[0]]
F[29,x,k[0]] = F[1,x]+F[30,x,k[0]]
F[30,x,k[0]] = F[31,x,k[0]]
F[31,x,k[0]] = F[29,x,k[0]]*F[32,x,k[0]]
F[32,x,k[0]] = k[0]*x
F[33,x,k[0]] = F[34,x]+F[36,x,k[0]]
F[34,x] = F[35,x]
F[35,x] = F[20,x]*F[8,x]
F[36,x,k[0]] = F[37,x,k[0]]
F[37,x,k[0]] = F[38,x,k[0]]*F[8,x]
F[38,x,k[0]] = F[39,x,k[0]]+F[44,x,k[0]]
F[39,x,k[0]] = F[40,x,k[0]]+F[43,x,k[0]]
F[40,x,k[0]] = F[41,x,k[0]]
F[41,x,k[0]] = F[32,x,k[0]]*F[42,x,k[0]]
F[42,x,k[0]] = F[13,x]+F[40,x,k[0]]
F[43,x,k[0]] = -k[0]*(F[24,x,1]-F[24,x,k[0]])/(-1+k[0])
F[44,x,k[0]] = F[45,x,k[0]]+F[47,x,k[0]]
F[45,x,k[0]] = F[46,x,k[0]]
F[46,x,k[0]] = F[13,x]^2*F[30,x,k[0]]
F[47,x,k[0]] = -k[0]*(F[36,x,1]-F[36,x,k[0]])/(-1+k[0])
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[13,x]^2
F[50,x] = F[33,x,1]
F[51,x] = F[52,x]
F[52,x] = F[10,x]*F[13,x]*F[53,x]
F[53,x] = F[13,x]+F[54,x]
F[54,x] = F[10,x]*F[2,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_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \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) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{48}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x , 1\right)
F_{23}\! \left(x , y\right) = F_{2}\! \left(x \right)+F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{25}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{26}\! \left(x , y\right) F_{32}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right)+F_{33}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{13}\! \left(x \right) F_{29}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{30}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{29}\! \left(x , y\right) F_{32}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = y x
F_{33}\! \left(x , y\right) = F_{34}\! \left(x \right)+F_{36}\! \left(x , y\right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{38}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{38}\! \left(x , y\right) = F_{39}\! \left(x , y\right)+F_{44}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)+F_{43}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{32}\! \left(x , y\right) F_{42}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{13}\! \left(x \right)+F_{40}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = -\frac{y \left(F_{24}\! \left(x , 1\right)-F_{24}\! \left(x , y\right)\right)}{-1+y}
F_{44}\! \left(x , y\right) = F_{45}\! \left(x , y\right)+F_{47}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{13} \left(x \right)^{2} F_{30}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = -\frac{y \left(F_{36}\! \left(x , 1\right)-F_{36}\! \left(x , y\right)\right)}{-1+y}
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13} \left(x \right)^{2}
F_{50}\! \left(x \right) = F_{33}\! \left(x , 1\right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{2}\! \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_4(x)*F_8(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_12(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_10(x)*F_8(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_13(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_8(x))
Eq(F_19(x), F_20(x) + F_51(x))
Eq(F_20(x), F_21(x) + F_48(x))
Eq(F_21(x), F_13(x) + F_22(x))
Eq(F_22(x), F_23(x, 1))
Eq(F_23(x, k_0), F_2(x) + F_24(x, k_0))
Eq(F_24(x, k_0), F_25(x, k_0))
Eq(F_25(x, k_0), F_26(x, k_0)*F_32(x, k_0))
Eq(F_26(x, k_0), F_23(x, k_0) + F_27(x, k_0))
Eq(F_27(x, k_0), F_28(x, k_0) + F_33(x, k_0))
Eq(F_28(x, k_0), F_13(x)*F_29(x, k_0))
Eq(F_29(x, k_0), F_1(x) + F_30(x, k_0))
Eq(F_30(x, k_0), F_31(x, k_0))
Eq(F_31(x, k_0), F_29(x, k_0)*F_32(x, k_0))
Eq(F_32(x, k_0), k_0*x)
Eq(F_33(x, k_0), F_34(x) + F_36(x, k_0))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_20(x)*F_8(x))
Eq(F_36(x, k_0), F_37(x, k_0))
Eq(F_37(x, k_0), F_38(x, k_0)*F_8(x))
Eq(F_38(x, k_0), F_39(x, k_0) + F_44(x, k_0))
Eq(F_39(x, k_0), F_40(x, k_0) + F_43(x, k_0))
Eq(F_40(x, k_0), F_41(x, k_0))
Eq(F_41(x, k_0), F_32(x, k_0)*F_42(x, k_0))
Eq(F_42(x, k_0), F_13(x) + F_40(x, k_0))
Eq(F_43(x, k_0), -F_24(x, 1) + (k_0*F_24(x, k_0) - F_24(x, 1))/(k_0 - 1))
Eq(F_44(x, k_0), F_45(x, k_0) + F_47(x, k_0))
Eq(F_45(x, k_0), F_46(x, k_0))
Eq(F_46(x, k_0), F_13(x)**2*F_30(x, k_0))
Eq(F_47(x, k_0), -F_36(x, 1) + (k_0*F_36(x, k_0) - F_36(x, 1))/(k_0 - 1))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_13(x)**2)
Eq(F_50(x), F_33(x, 1))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_10(x)*F_13(x)*F_53(x))
Eq(F_53(x), F_13(x) + F_54(x))
Eq(F_54(x), F_10(x)*F_2(x))
Pack JSON:
{"expansion_strats": [[{"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.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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