0132_0213_0231_0312_1032_1302_3120
Counting sequence:
1, 1, 2, 6, 17, 43, 104, 242, 549, 1225, 2702, 5908, 12827, 27683, 59436, 127030, 270401, 573517, 1212506, 2556008, 5374071, 11272327, 23593112, 49283242, 102760637, 213909713, 444596454, 922747132, 1912602899, 3959423275, 8187281732, 16911434078, 34896609657, 71940702613, 148176372146, 304942678480, 627065225711, 1288490189327, 2645699854896, 5428838662738, 11132555231861, 22814866277017, 46729244181182, 95657511617252, 195713069744907, 400222232511283, 818036651066204, 1671257674220422, 3412884092617649, 6966505673589725, 14214486323889162, 28991922601198648, 59109745109238887, 120471290032161943, 245446179691693256, 499899558638126330, 1017813515785733421, 2071655828590429537, 4215369251218785686, 8574853690513425868, 17437937757178562051, 35452336266660546107, 72057594037927937652, 146421031085069567662, 297453748188566521577, 604130868413987817253, 1226708480901685184354, 2490310449950789470112, 5054407876196417144799, 10256389704982510700575, 20807927315144374224992, 42206150440647454099618, 85592892502012319500517, 173546968245459461605673, 351816302973788568422766, 713077338913316427270580, 1445044143758111435393531, 2927867219379180032494147, 5931292302484274388404876, 12013700332420377423645398, 24329632119744412140964641, 49263727149296138869279597, 99736380118206906913262522, 201890611875643072175934472, 408616927029744661050690647, 826905260616406355499027623, 1673153334346646777793350904, 3384992294920961689177296202, 6847355842297259645535784349, 13849454189505191825433955825, 28008393388831728719592689222, 56635756797306147576634936988, 114509453633897675428168994547, 231494787346366111406136233803, 467941334849873743911868960676, 945786190014030530022930911230, 1911379420656627144444247806041, 3862372922570386457685267583157, 7803974007655037252964079112466, 15766404340338603181115246121328, 31849721330734263712604668039631
Generating function in Maple syntax:
-(4*x^8-6*x^7+4*x^6+4*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{8}-6 x^{7}+4 x^{6}+4 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**8 + 6*x**7 - 4*x**6 - 4*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+4*x^8-6*x^7+4*x^6+4*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{8}-6 x^{7}+4 x^{6}+4 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,1/8*(2*n+1)*2^n+1/2*n^2-9/2*n+9)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(2 n +1\right) 2^{n}}{8}+\frac{n^{2}}{2}-\frac{9 n}{2}+9 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 43
a(6) = 104
a(7) = 242
a(8) = 549
a(n+2) = -4*a(n)+4*a(n+1)+1/2*(n-4)*(n-9), n >= 9
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) = 17
a \! \left(5\right) = 43
a \! \left(6\right) = 104
a \! \left(7\right) = 242
a \! \left(8\right) = 549
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(n -4\right) \left(n -9\right)}{2}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22780/
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[16,x]*F[4,x]
F[4,x] = F[35,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[21,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[29,x]
F[23,x] = 0
F[24,x] = F[16,x]*F[25,x]
F[25,x] = F[14,x]+F[26,x]
F[26,x] = F[23,x]+F[24,x]+F[27,x]
F[27,x] = F[16,x]*F[28,x]
F[28,x] = F[18,x]+F[26,x]
F[29,x] = F[16,x]*F[30,x]
F[30,x] = F[28,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[16,x]*F[34,x]
F[34,x] = F[18,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[16,x]*F[37,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[39,x]+F[45,x]
F[39,x] = F[40,x]*F[5,x]
F[40,x] = F[20,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[16,x]*F[20,x]*F[43,x]
F[43,x] = F[1,x]+F[44,x]
F[44,x] = F[41,x]
F[45,x] = F[46,x]
F[46,x] = F[13,x]*F[16,x]*F[41,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[16,x]*F[20,x]*F[49,x]
F[49,x] = F[1,x]+F[50,x]
F[50,x] = F[47,x]
F[51,x] = F[41,x]*F[52,x]
F[52,x] = F[53,x]+F[6,x]
F[53,x] = F[20,x]+F[47,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{35}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{21}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{5}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{41}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{1}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{41}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{6}\! \left(x \right)
F_{53}\! \left(x \right) = F_{20}\! \left(x \right)+F_{47}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_35(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_16(x)*F_8(x))
Eq(F_8(x), F_21(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_29(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_16(x)*F_25(x))
Eq(F_25(x), F_14(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_24(x) + F_27(x))
Eq(F_27(x), F_16(x)*F_28(x))
Eq(F_28(x), F_18(x) + F_26(x))
Eq(F_29(x), F_16(x)*F_30(x))
Eq(F_30(x), F_28(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_16(x)*F_34(x))
Eq(F_34(x), F_18(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_16(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_39(x) + F_45(x))
Eq(F_39(x), F_40(x)*F_5(x))
Eq(F_40(x), F_20(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_16(x)*F_20(x)*F_43(x))
Eq(F_43(x), F_1(x) + F_44(x))
Eq(F_44(x), F_41(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_13(x)*F_16(x)*F_41(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_16(x)*F_20(x)*F_49(x))
Eq(F_49(x), F_1(x) + F_50(x))
Eq(F_50(x), F_47(x))
Eq(F_51(x), F_41(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_6(x))
Eq(F_53(x), F_20(x) + F_47(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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