0132_0231_1320_3120
Counting sequence:
1, 1, 2, 6, 20, 65, 200, 584, 1632, 4400, 11520, 29440, 73728, 181504, 440320, 1054720, 2498560, 5861376, 13631488, 31457280, 72089600, 164167680, 371720192, 837287936, 1876951040, 4189061120, 9311354880, 20619198464, 45499809792, 100076093440, 219445985280, 479828377600, 1046361407488, 2276064231424, 4939212390400, 10694468567040, 23106924052480, 49825915600896, 107236743446528, 230382045757440, 494093037731840, 1057936344350720, 2261695418335232, 4827955557564416, 10291428835983360, 21907769183436800, 46575312552591360, 98894473848684544, 209734042021527552, 444290658551726080, 940126422213591040, 1987213335577231360, 4196228952802459648, 8852106542573420544, 18656161456382279680, 39282647749739151360, 82641053162248601600, 173708341227307401216, 364827598614029139968, 765611936652984320000, 1605443195165034414080, 3364008777660665692160, 7043773932395531599872, 14738372054141628317696, 30817591818141019668480, 64396430639815437844480, 134476764297342631280640, 280648764337417118285824, 585352082946951491878912, 1220159886755518291640320, 2541961333357176212684800, 5292739809628744549662720, 11014329525483088755294208, 22909085252116186990116864, 47625065979740371980124160, 98957189648533415452999680, 205517389334486959700049920, 426623310428926618250510336, 885198152480949256016887808, 1835867193880403273276456960, 3805849595874304470621880320, 7886352026392307506872320000, 16335005674632869408629850112, 33820908729538865791579979776, 69996804955687029215487590400, 144811179302538355692179292160, 299475104034935939158213918720, 619095747927930058883731554304, 1279372344975212845974654287872, 2642905269808717716575653724160, 5457768148199420286961513922560, 11266801782546067226925654343680, 23250989817858012323155351175168, 47966772187200450856585091088384, 98923788539294737767189016739840, 203950621472266400289630715904000, 420354919739743726144746621501440, 866117320845780597051522868576256, 1784059763496204353931422673666048, 3673809895786437334212632977080320, 7563120393611673666679705499074560
Generating function in Maple syntax:
(x^5+4*x^4-18*x^3+18*x^2-7*x+1)/(2*x-1)^4
Generating function in latex syntax:
\frac{x^{5}+4 x^{4}-18 x^{3}+18 x^{2}-7 x +1}{\left(2 x -1\right)^{4}}
Generating function in sympy syntax:
(x**5 + 4*x**4 - 18*x**3 + 18*x**2 - 7*x + 1)/(2*x - 1)**4
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^4*F(x)+x^5+4*x^4-18*x^3+18*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{4} F \! \left(x \right)+x^{5}+4 x^{4}-18 x^{3}+18 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/192*2^n*(n^3+15*n^2-46*n+120))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{2^{n} \left(n^{3}+15 n^{2}-46 n +120\right)}{192} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+4) = -16*a(n)+32*a(n+1)-24*a(n+2)+8*a(n+3), n >= 6
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) = 65
a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21246/
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[10,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[10,x]*F[24,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[10,x]*F[12,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[13,x]+F[19,x]
F[23,x] = F[10,x]*F[15,x]
F[24,x] = F[25,x]+F[70,x]
F[25,x] = F[26,x]+F[52,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[10,x]*F[12,x]*F[30,x]
F[33,x] = F[16,x]+F[34,x]
F[34,x] = F[20,x]+F[35,x]+F[37,x]
F[35,x] = F[10,x]*F[36,x]
F[36,x] = F[31,x]+F[34,x]
F[37,x] = F[10,x]*F[38,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[16,x]+F[40,x]
F[40,x] = F[20,x]+F[41,x]+F[43,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[13,x]+F[40,x]
F[43,x] = F[10,x]*F[39,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[46,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[16,x]+F[45,x]
F[48,x] = 2*F[20,x]+F[49,x]+F[51,x]
F[49,x] = F[10,x]*F[50,x]
F[50,x] = F[40,x]+F[48,x]
F[51,x] = F[10,x]*F[44,x]
F[52,x] = F[16,x]+F[53,x]
F[53,x] = F[20,x]+F[35,x]+F[54,x]
F[54,x] = F[10,x]*F[55,x]
F[55,x] = F[33,x]+F[56,x]
F[56,x] = F[45,x]+F[57,x]
F[57,x] = 2*F[20,x]+F[58,x]+F[60,x]
F[58,x] = F[10,x]*F[59,x]
F[59,x] = F[34,x]+F[57,x]
F[60,x] = F[10,x]*F[61,x]
F[61,x] = F[44,x]+F[62,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[64,x]
F[64,x] = F[10,x]*F[65,x]
F[65,x] = F[45,x]+F[63,x]
F[66,x] = 3*F[20,x]+F[67,x]+F[69,x]
F[67,x] = F[10,x]*F[68,x]
F[68,x] = F[48,x]+F[66,x]
F[69,x] = F[10,x]*F[62,x]
F[70,x] = F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[18,x]^2*F[31,x]
F[73,x] = F[74,x]
F[74,x] = F[10,x]*F[16,x]*F[31,x]*F[75,x]
F[75,x] = F[18,x]+F[42,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{10}\! \left(x \right) F_{24}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{70}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{52}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{33}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{45}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{44}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{45}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = 3 F_{20}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{10}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{48}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{10}\! \left(x \right) F_{62}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{31}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{31}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_10(x)*F_24(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_10(x)*F_12(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_13(x) + F_19(x))
Eq(F_23(x), F_10(x)*F_15(x))
Eq(F_24(x), F_25(x) + F_70(x))
Eq(F_25(x), F_26(x) + F_52(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_10(x)*F_12(x)*F_30(x))
Eq(F_33(x), F_16(x) + F_34(x))
Eq(F_34(x), F_20(x) + F_35(x) + F_37(x))
Eq(F_35(x), F_10(x)*F_36(x))
Eq(F_36(x), F_31(x) + F_34(x))
Eq(F_37(x), F_10(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_16(x) + F_40(x))
Eq(F_40(x), F_20(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_40(x))
Eq(F_43(x), F_10(x)*F_39(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_16(x) + F_45(x))
Eq(F_48(x), 2*F_20(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_10(x)*F_50(x))
Eq(F_50(x), F_40(x) + F_48(x))
Eq(F_51(x), F_10(x)*F_44(x))
Eq(F_52(x), F_16(x) + F_53(x))
Eq(F_53(x), F_20(x) + F_35(x) + F_54(x))
Eq(F_54(x), F_10(x)*F_55(x))
Eq(F_55(x), F_33(x) + F_56(x))
Eq(F_56(x), F_45(x) + F_57(x))
Eq(F_57(x), 2*F_20(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_10(x)*F_59(x))
Eq(F_59(x), F_34(x) + F_57(x))
Eq(F_60(x), F_10(x)*F_61(x))
Eq(F_61(x), F_44(x) + F_62(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_10(x)*F_65(x))
Eq(F_65(x), F_45(x) + F_63(x))
Eq(F_66(x), 3*F_20(x) + F_67(x) + F_69(x))
Eq(F_67(x), F_10(x)*F_68(x))
Eq(F_68(x), F_48(x) + F_66(x))
Eq(F_69(x), F_10(x)*F_62(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_18(x)**2*F_31(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_10(x)*F_16(x)*F_31(x)*F_75(x))
Eq(F_75(x), F_18(x) + F_42(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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