0132_0213_1032_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 419, 1100, 2848, 7315, 18712, 47796, 122115, 312404, 800760, 2057107, 5296832, 13669548, 35351363, 91598108, 237739024, 617948947, 1608243496, 4190004324, 10926117891, 28512851876, 74453054184, 194510766355, 508374601808, 1329136644060, 3475948322627, 9092265872876, 23787427523200, 62242099409299, 162880888646200, 426280307445012, 1115710925585667, 2920337073236468, 7644235142234328, 20010169330210963, 52381737362933984, 137125696909754892, 358976111912844611, 939763056410179004, 2460231693457236976, 6440764898194109971, 16861719953497226824, 44143691274855793860, 115567911311814512643, 302557087173332282180, 792097298496183054792, 2073722423416241613331, 5429044639004637826160, 14213359702201957104444, 37210928633009990280515, 97419210024045899953292, 255046260086364227270752, 667718669515121307759763, 1748107910990351728846168, 4576601316461043906526068, 11981688400287811970370819, 31368448319962079812152212, 82123624854257050777793976, 215002361679204814537799059, 562883328750305867655047936, 1473647357093923719638846316, 3858058198352515116829718339, 10100526131158977208277211740, 26443517944621639515436619728, 69230023127913411058063846675, 181246542141959580509140905832, 474509584408499398990778016036, 1242282172714310943147325780995, 3252336855815386463102053300964, 8514728236532571270025719463080, 22291847532661406511839230556947, 58360813709765073629480932717712, 152790592274371199173102907761308, 400010960431044361619869309186883, 1047242283578595697420673733621548, 2741715879273294626658660672483904, 7177905331876160519684668551765907, 18792000071020468696846750931332216, 49198094789306883280143766964561364, 128802284110725604922931657057600771, 337208757165685075768886901699914036, 882823986622288504387283409627838104, 2311263201153755388286237983559694995, 6050965613705441936030311720212838688, 15841633633618127718467123268220809420, 41473935274305313311785237732371175747, 108580172163301071391895604156011109116, 284266581162985449194272913052449372080, 744219571219192432812380431360012323859, 1948392132279190282407212213195139984520, 5100956825182623520580802351460915902852, 13354478342387266971364004085459431276547, 34962478200196543736940262309062639044228, 91532956254597482845057755481222236115848, 239636390556306913846920685075998025872403
Generating function in Maple syntax:
-(2*x^6-2*x^5+6*x^4-19*x^3+18*x^2-7*x+1)/(x-1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{2 x^{6}-2 x^{5}+6 x^{4}-19 x^{3}+18 x^{2}-7 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**6 + 2*x**5 - 6*x**4 + 19*x**3 - 18*x**2 + 7*x - 1)/((x - 1)*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)*(2*x-1)^2*F(x)+2*x^6-2*x^5+6*x^4-19*x^3+18*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+2 x^{6}-2 x^{5}+6 x^{4}-19 x^{3}+18 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n)+1+1/4*(n-4)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}+1+\frac{\left(n -4\right) 2^{n}}{4} & \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) = 156
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1, 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) = 56
a \! \left(6\right) = 156
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18354/
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[12,x]*F[4,x]
F[4,x] = F[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,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[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[6,x]
F[17,x] = F[18,x]+F[19,x]+F[40,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[29,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[18,x]+F[23,x]+F[26,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[23,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[26,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[18,x]+F[19,x]+F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[30,x]+F[6,x]
F[33,x] = 2*F[18,x]+F[34,x]+F[37,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[30,x]+F[36,x]
F[36,x] = F[34,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[14,x]+F[39,x]
F[39,x] = F[37,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[32,x]+F[42,x]
F[42,x] = F[43,x]+F[54,x]
F[43,x] = F[18,x]+F[44,x]+F[49,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[24,x]+F[46,x]
F[46,x] = F[14,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[46,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[13,x]+F[51,x]
F[51,x] = F[43,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[51,x]
F[54,x] = F[55,x]
F[55,x] = F[12,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{16}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{40}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{54}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{24}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_16(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_12(x)*F_8(x))
Eq(F_8(x), F_13(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_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_6(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_40(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_29(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_26(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_23(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_26(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_18(x) + F_19(x) + F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_30(x) + F_6(x))
Eq(F_33(x), 2*F_18(x) + F_34(x) + F_37(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_30(x) + F_36(x))
Eq(F_36(x), F_34(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_14(x) + F_39(x))
Eq(F_39(x), F_37(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_32(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_54(x))
Eq(F_43(x), F_18(x) + F_44(x) + F_49(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_24(x) + F_46(x))
Eq(F_46(x), F_14(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_46(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_13(x) + F_51(x))
Eq(F_51(x), F_43(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_51(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(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": 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:
{"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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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