0132_0213_2103_2130_3102
Counting sequence:
1, 1, 2, 6, 19, 56, 155, 412, 1068, 2726, 6893, 17337, 43495, 109062, 273718, 688296, 1735375, 4388819, 11136185, 28352104, 72421008, 185568602, 476879249, 1228750701, 3173590219, 8213901066, 21298243690, 55312779852, 143845275283, 374511769991, 976015824413, 2545667188972, 6644174972100, 17351021593982, 45332654140373, 118485074247201, 309779508088495, 810127600545294, 2119069990222750, 5543848259749104, 14505662970893143, 37958812642030907, 99340675824389249, 260000061632016496, 680527155359466744, 1781304327516184226, 4662806384561248025, 12205904263865378517, 31952379729314258323, 83645966064357093906, 218974540939665307858, 573254804504967133332, 1500742338488544080155, 3928873413243839666927, 10285672705984952852453, 26927718833071255667956, 70496600524751896112940, 184560252027935906564294, 483180363528169577622365, 1264972988782422319528329, 3311722362838841082954103, 8670160520895279254915862, 22698689808383938157191174, 59425765581701993777993016, 155578311068240923444682719, 407308557151084087230581411, 1066346101394728307570163785, 2791727151806793965467367032, 7308830006775715222275662688, 19134751855814139696757306346, 50095402889618237278957320353, 131151410161224809728658617917, 343358731634093520469931027035, 898924587434681139283770378522, 2353414625136728180950597382458, 6161318454772967082254994377436, 16130539027914618835926701436835, 42230295115530226170509070943319, 110560339107622440333652270079405, 289450707412162903999849284785404, 757791752782939252745555441066388, 1983924488714754075946371702755342, 5193981585819649005750181735698149, 13598020007389542313366528797547761, 35600077900945955572520508908947519, 93202212598952606013726336470793246, 244006557650936615444292123074063470, 638817455758730199963944539693222272, 1672445800222228959484955775140966503, 4378519925671529036782943357805004811, 11463113937449076277325382436636256273, 30010821806228940054757320150455988352, 78569351316785146793129342121101175240, 205697231808025779658649961577735310642, 538522343420544955389255835111815593513, 1409869797050713536988960347228278943141, 3691087044866383434651612560318998219555, 9663391331697932141303170154555454632994, 25299086938283767490232549011120615166370, 66233869458775855075786768505492836647204, 173402521388298415198484035538811486958123
Generating function in Maple syntax:
-(x^4-x^3+4*x^2-4*x+1)*(x^5-10*x^3+13*x^2-6*x+1)/(x^2-3*x+1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
-\frac{\left(x^{4}-x^{3}+4 x^{2}-4 x +1\right) \left(x^{5}-10 x^{3}+13 x^{2}-6 x +1\right)}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(-x**4 + x**3 - 4*x**2 + 4*x - 1)*(x**5 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**2*(2*x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*(2*x-1)^3*F(x)+(x^4-x^3+4*x^2-4*x+1)*(x^5-10*x^3+13*x^2-6*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)+\left(x^{4}-x^{3}+4 x^{2}-4 x +1\right) \left(x^{5}-10 x^{3}+13 x^{2}-6 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/320*(-32*5^(1/2)+160)*(3/2-1/2*5^(1/2))^(-n)+1/320*(32*5^(1/2)+160)*(3/2+1/2*5^(1/2))^(-n)+1/320*(5*n^2+45*n-100)*2^n-n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-32 \sqrt{5}+160\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{320}+\frac{\left(32 \sqrt{5}+160\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{320}+\\\frac{\left(5 n^{2}+45 n -100\right) 2^{n}}{320}-n +2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 155
a(7) = 412
a(8) = 1068
a(9) = 2726
a(n+5) = 8*a(n)-36*a(n+1)+50*a(n+2)-31*a(n+3)+9*a(n+4)-n+4, n >= 10
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) = 56
a \! \left(6\right) = 155
a \! \left(7\right) = 412
a \! \left(8\right) = 1068
a \! \left(9\right) = 2726
a \! \left(n +5\right) = 8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)-n +4, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18563/
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[5,x]
F[4,x] = x
F[5,x] = F[16,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[30,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[6,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[17,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[21,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]+F[46,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[31,x]+F[36,x]+F[41,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[35,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[43,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[55,x]
F[48,x] = F[17,x]+F[49,x]
F[49,x] = F[31,x]+F[50,x]+F[54,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[11,x]+F[53,x]
F[53,x] = F[15,x]+F[31,x]+F[36,x]
F[54,x] = F[4,x]*F[48,x]
F[55,x] = F[56,x]+F[62,x]
F[56,x] = F[31,x]+F[32,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[24,x]+F[59,x]
F[59,x] = F[56,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[59,x]
F[62,x] = 2*F[31,x]+F[63,x]+F[67,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[14,x]+F[66,x]
F[66,x] = F[45,x]
F[67,x] = F[4,x]*F[55,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{16}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{46}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{31}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{62}\! \left(x \right)
F_{56}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{24}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{14}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{45}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_16(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_30(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_6(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_17(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_21(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_46(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_41(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_35(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_43(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_55(x))
Eq(F_48(x), F_17(x) + F_49(x))
Eq(F_49(x), F_31(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_11(x) + F_53(x))
Eq(F_53(x), F_15(x) + F_31(x) + F_36(x))
Eq(F_54(x), F_4(x)*F_48(x))
Eq(F_55(x), F_56(x) + F_62(x))
Eq(F_56(x), F_31(x) + F_32(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_24(x) + F_59(x))
Eq(F_59(x), F_56(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_59(x))
Eq(F_62(x), 2*F_31(x) + F_63(x) + F_67(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_14(x) + F_66(x))
Eq(F_66(x), F_45(x))
Eq(F_67(x), F_4(x)*F_55(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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "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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