0132_1203_1320_2013
Counting sequence:
1, 1, 2, 6, 20, 64, 194, 563, 1579, 4311, 11524, 30297, 78601, 201734, 513185, 1295809, 3251392, 8114206, 20154966, 49857538, 122885702, 301903459, 739567951, 1806992865, 4404640440, 10713537411, 26007847503, 63022504554, 152464532673, 368280696689, 888331353342, 2139935327606, 5148670969848, 12373534166276, 29704921696810, 71240671037795, 170694222754739, 408624923681499, 977390289533708, 2335976290152157, 5578848443422741, 13314149925196526, 31753440425447937, 75681745065982961, 180271874254673500, 429154643858107214, 1021081275764533274, 2428165685114000518, 5771357337050320494, 13710987289895021011, 32558131892844389559, 77278472915195571877, 183346953036914925056, 434822807587905641159, 1030816349754363695515, 2442795753529744607314, 5786767510001207225857, 13703570424035062914081, 32440366243382178735226, 76771012143986258758118, 181623989200792652263132, 429554474786409507171784, 1015633010876277443301906, 2400673514955069262707795, 5672978385673178247392955, 13402156968686142553959711, 31653915808918578728963988, 74743394079184122274767009, 176446933712981423061105953, 416441970914231269187427414, 982645695725092431855731297, 2318165189922434964167540961, 5467635011285649619263767288, 12893326970503966105480192126, 30397883327917460415929042270, 71653428421714086610584586250, 168868329805521210965329417494, 397904881091869088135469082499, 937415836574991609455082853663, 2208052104656973473936401008937, 5200107620056843930972011238856, 12244561151731970832926056527051, 28827290143662521577055525558247, 67857189988893686278223428084666, 159705940571481391805153936423105, 375821002442796425213487489929489, 884254682622277889339228718548854, 2080230931512034546225251816992598, 4893110813278842974414661242382784, 11507984874597566825116413311389836, 27061732959105851346117917993065594, 63629034183696510661890301518024707, 149589025788806362574853827683347011, 351633335347893124790469158307319459, 826470654498680095401051223285272220, 1942285533637619758118614112763964069, 4564025534100997621117492001423281197, 10723424145791047898510519094744134206, 25192448172975960112772826701350855361, 59177926566256734367272694603551453073, 138996087328969418042635173043703249812
Generating function in Maple syntax:
-(2*x^7-9*x^6+19*x^5-29*x^4+30*x^3-20*x^2+7*x-1)/(x-1)^2/(x^3-2*x^2+3*x-1)^2
Generating function in latex syntax:
-\frac{2 x^{7}-9 x^{6}+19 x^{5}-29 x^{4}+30 x^{3}-20 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(x^{3}-2 x^{2}+3 x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**7 + 9*x**6 - 19*x**5 + 29*x**4 - 30*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**2*(x**3 - 2*x**2 + 3*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(x^3-2*x^2+3*x-1)^2*F(x)+2*x^7-9*x^6+19*x^5-29*x^4+30*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+2 x^{7}-9 x^{6}+19 x^{5}-29 x^{4}+30 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
1/317400*(-5865*2^(2/3)*(-3/17*(n-383/69)*(I-1/3*3^(1/2))*23^(1/2)+(I*3^(1/2)-1)*(n-39/17))*(11+3*23^(1/2)*3^(1/2))^(1/3)-138*2^(1/3)*(6*(I+1/3*3^(1/2))*(n-193/138)*23^(1/2)+(n-36)*(1+I*3^(1/2)))*(11+3*23^(1/2)*3^(1/2))^(2/3)+69000*n)*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(5865*2^(2/3)*(-3/17*(n-383/69)*(I+1/3*3^(1/2))*23^(1/2)+(n-39/17)*(1+I*3^(1/2)))*(11+3*23^(1/2)*3^(1/2))^(1/3)+138*(6*(I-1/3*3^(1/2))*(n-193/138)*23^(1/2)+(I*3^(1/2)-1)*(n-36))*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+69000*n)*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(690*(3^(1/2)*(n-383/69)*23^(1/2)-17*n+39)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+552*2^(1/3)*(3^(1/2)*(n-193/138)*23^(1/2)+1/2*n-18)*(11+3*23^(1/2)*3^(1/2))^(2/3)+69000*n)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+n+1
Explicit closed form in latex syntax:
\frac{\left(-5865 \,2^{\frac{2}{3}} \left(-\frac{3 \left(n -\frac{383}{69}\right) \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{23}}{17}+\left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{39}{17}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-138 \,2^{\frac{1}{3}} \left(6 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{193}{138}\right) \sqrt{23}+\left(n -36\right) \left(1+\mathrm{I} \sqrt{3}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+69000 n \right) \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}+\frac{\left(5865 \,2^{\frac{2}{3}} \left(-\frac{3 \left(n -\frac{383}{69}\right) \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{23}}{17}+\left(n -\frac{39}{17}\right) \left(1+\mathrm{I} \sqrt{3}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+138 \left(6 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{193}{138}\right) \sqrt{23}+\left(\mathrm{I} \sqrt{3}-1\right) \left(n -36\right)\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+69000 n \right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}+\frac{\left(690 \left(\sqrt{3}\, \left(n -\frac{383}{69}\right) \sqrt{23}-17 n +39\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+552 \,2^{\frac{1}{3}} \left(\sqrt{3}\, \left(n -\frac{193}{138}\right) \sqrt{23}+\frac{n}{2}-18\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+69000 n \right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400}+n +1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 194
a(7) = 563
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5)+n+3, 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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 194
a \! \left(7\right) = 563
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)+n +3, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15561/
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[23,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[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[11,x]
F[19,x] = F[14,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[20,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[47,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[21,x]
F[31,x] = F[24,x]+F[32,x]
F[32,x] = 2*F[15,x]+F[33,x]+F[37,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[28,x]
F[36,x] = F[32,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[7,x]
F[44,x] = F[40,x]
F[45,x] = F[46,x]
F[46,x] = F[39,x]*F[4,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[40,x]+F[53,x]
F[53,x] = F[46,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_{23}\! \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_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{40}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{46}\! \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_23(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) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_11(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_20(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_47(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_21(x))
Eq(F_31(x), F_24(x) + F_32(x))
Eq(F_32(x), 2*F_15(x) + F_33(x) + F_37(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_7(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_39(x)*F_4(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_40(x) + F_53(x))
Eq(F_53(x), F_46(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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