0132_0213_1302_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 50, 113, 231, 444, 825, 1512, 2772, 5129, 9620, 18307, 35301, 68810, 135235, 267362, 530746, 1056479, 2106726, 4205797, 8402291, 16793384, 33573405, 67130988, 134243376, 268465029, 536904840, 1073780567, 2147527697, 4295017174, 8589990855, 17179932422, 34359809206, 68719555835, 137439041530, 274878004697, 549755922111, 1099511747284, 2199023387201, 4398046655792, 8796093180876, 17592186218049, 35184372278460, 70368744384363, 140737488580221, 281474976954914, 562949953686155, 1125899907129322, 2251799813995122, 4503599627704919, 9007199255101390, 18014398509869837, 36028797019380811, 72057594038375360, 144115188076335525, 288230376152225332, 576460752303972776, 1152921504607433789, 2305843009214320176, 4611686018428055487, 9223372036855486761, 18446744073710308014, 36893488147419907215, 73786976294839060238, 147573952589677318766, 295147905179353786099, 590295810358706668770, 1180591620717412379777, 2361183241434823745047, 4722366482869646416364, 9444732965739291697225, 18889465931478582194552, 37778931862957163122116, 75557863725914324907385, 151115727451828648405220, 302231454903657295325267, 604462909807314589086741, 1208925819614629176527994, 2417851639229258351325651, 4835703278458516700832882, 9671406556917033399755946, 19342813113834066797507279, 38685626227668133592911670, 77371252455336267183618613, 154742504910672534364927011, 309485009821345068727434584, 618970019642690137452336685, 1237940039285380274902023932, 2475880078570760549801277472, 4951760157141521099599659509, 9903520314283042199196294360, 19807040628566084398389430567, 39614081257132168796775565121, 79228162514264337593547691910, 158456325028528675187091798615, 316912650057057350374179860502, 633825300114114700748355828006, 1267650600228229401496707601899
Generating function in Maple syntax:
(4*x^6-9*x^5+12*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{4 x^{6}-9 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(4*x**6 - 9*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)-4*x^6+9*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)-4 x^{6}+9 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-2+2^n+1/4*n^3-49/24*n^2+11/4*n+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -2+2^{n}+\frac{n^{3}}{4}-\frac{49 n^{2}}{24}+\frac{11 n}{4}+\frac{n^{4}}{24} & \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) = 50
a(6) = 113
a(n+1) = 2*a(n)-1/24*(n-1)*(n^3+3*n^2-70*n+72), n >= 7
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) = 50
a \! \left(6\right) = 113
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{3}+3 n^{2}-70 n +72\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21636/
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[39,x]+F[5,x]
F[5,x] = F[1,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[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[11,x]+F[22,x]
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[28,x]+F[33,x]
F[28,x] = F[19,x]+F[29,x]
F[29,x] = F[23,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[15,x]+F[29,x]
F[32,x] = F[28,x]*F[8,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[19,x]+F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[33,x]*F[8,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]*F[8,x]
F[41,x] = F[42,x]+F[51,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]+F[48,x]
F[44,x] = F[14,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[21,x]^2
F[47,x] = F[30,x]
F[48,x] = F[21,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[15,x]*F[21,x]*F[8,x]
F[51,x] = F[21,x]*F[52,x]
F[52,x] = F[53,x]+F[58,x]
F[53,x] = F[19,x]+F[54,x]
F[54,x] = F[23,x]+F[55,x]+F[57,x]
F[55,x] = F[56,x]*F[8,x]
F[56,x] = F[15,x]+F[54,x]
F[57,x] = F[53,x]*F[8,x]
F[58,x] = F[59,x]+F[70,x]
F[59,x] = F[23,x]+F[60,x]+F[63,x]
F[60,x] = F[61,x]*F[8,x]
F[61,x] = F[11,x]+F[62,x]
F[62,x] = F[60,x]
F[63,x] = F[64,x]*F[8,x]
F[64,x] = F[65,x]+F[69,x]
F[65,x] = F[19,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[68,x]*F[8,x]
F[68,x] = F[15,x]+F[66,x]
F[69,x] = F[59,x]
F[70,x] = F[71,x]
F[71,x] = F[72,x]*F[8,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[74,x]
F[74,x] = F[75,x]*F[8,x]
F[75,x] = F[19,x]+F[73,x]
F[76,x] = F[71,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_{39}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \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_{18}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{51}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21} \left(x \right)^{2}
F_{47}\! \left(x \right) = F_{30}\! \left(x \right)
F_{48}\! \left(x \right) = F_{21}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{19}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{23}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right) F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right) F_{8}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{70}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right)+F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right) F_{8}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right) F_{8}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{19}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right) F_{8}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{59}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right) F_{8}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right) F_{8}\! \left(x \right)
F_{75}\! \left(x \right) = F_{19}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{71}\! \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_39(x) + F_5(x))
Eq(F_5(x), F_1(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_18(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_11(x) + F_22(x))
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_28(x) + F_33(x))
Eq(F_28(x), F_19(x) + F_29(x))
Eq(F_29(x), F_23(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_15(x) + F_29(x))
Eq(F_32(x), F_28(x)*F_8(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_19(x) + F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_33(x)*F_8(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x)*F_8(x))
Eq(F_41(x), F_42(x) + F_51(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x) + F_48(x))
Eq(F_44(x), F_14(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_21(x)**2)
Eq(F_47(x), F_30(x))
Eq(F_48(x), F_21(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_15(x)*F_21(x)*F_8(x))
Eq(F_51(x), F_21(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_58(x))
Eq(F_53(x), F_19(x) + F_54(x))
Eq(F_54(x), F_23(x) + F_55(x) + F_57(x))
Eq(F_55(x), F_56(x)*F_8(x))
Eq(F_56(x), F_15(x) + F_54(x))
Eq(F_57(x), F_53(x)*F_8(x))
Eq(F_58(x), F_59(x) + F_70(x))
Eq(F_59(x), F_23(x) + F_60(x) + F_63(x))
Eq(F_60(x), F_61(x)*F_8(x))
Eq(F_61(x), F_11(x) + F_62(x))
Eq(F_62(x), F_60(x))
Eq(F_63(x), F_64(x)*F_8(x))
Eq(F_64(x), F_65(x) + F_69(x))
Eq(F_65(x), F_19(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_68(x)*F_8(x))
Eq(F_68(x), F_15(x) + F_66(x))
Eq(F_69(x), F_59(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_72(x)*F_8(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_75(x)*F_8(x))
Eq(F_75(x), F_19(x) + F_73(x))
Eq(F_76(x), F_71(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": 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:
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