0213_0321_1230_2130_2301
Counting sequence:
1, 1, 2, 6, 19, 49, 112, 241, 502, 1027, 2080, 4189, 8410, 16855, 33748, 67537, 135118, 270283, 540616, 1081285, 2162626, 4325311, 8650684, 17301433, 34602934, 69205939, 138411952, 276823981, 553648042, 1107296167, 2214592420, 4429184929, 8858369950, 17716739995, 35433480088, 70866960277, 141733920658, 283467841423, 566935682956, 1133871366025, 2267742732166, 4535485464451, 9070970929024, 18141941858173, 36283883716474, 72567767433079, 145135534866292, 290271069732721, 580542139465582, 1161084278931307, 2322168557862760, 4644337115725669, 9288674231451490, 18577348462903135, 37154696925806428, 74309393851613017, 148618787703226198, 297237575406452563, 594475150812905296, 1188950301625810765, 2377900603251621706, 4755801206503243591, 9511602413006487364, 19023204826012974913, 38046409652025950014, 76092819304051900219, 152185638608103800632, 304371277216207601461, 608742554432415203122, 1217485108864830406447, 2434970217729660813100, 4869940435459321626409, 9739880870918643253030, 19479761741837286506275, 38959523483674573012768, 77919046967349146025757, 155838093934698292051738, 311676187869396584103703, 623352375738793168207636, 1246704751477586336415505, 2493409502955172672831246, 4986819005910345345662731, 9973638011820690691325704, 19947276023641381382651653, 39894552047282762765303554, 79789104094565525530607359, 159578208189131051061214972, 319156416378262102122430201, 638312832756524204244860662, 1276625665513048408489721587, 2553251331026096816979443440, 5106502662052193633958887149, 10213005324104387267917774570, 20426010648208774535835549415, 40852021296417549071671099108, 81704042592835098143342198497, 163408085185670196286684397278, 326816170371340392573368794843, 653632340742680785146737589976, 1307264681485361570293475180245, 2614529362970723140586950360786
Generating function in Maple syntax:
(x^6+x^5-3*x^4-x^3-3*x^2+3*x-1)/(2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{6}+x^{5}-3 x^{4}-x^{3}-3 x^{2}+3 x -1}{\left(2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**6 + x**5 - 3*x**4 - x**3 - 3*x**2 + 3*x - 1)/((x - 1)**2*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^2*F(x)+x^6+x^5-3*x^4-x^3-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}+x^{5}-3 x^{4}-x^{3}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,-2-3*n+33/16*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ -2-3 n +\frac{33 \,2^{n}}{16} & \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) = 49
a(6) = 112
a(n+1) = 2*a(n)+3*n-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) = 49
a \! \left(6\right) = 112
a \! \left(n +1\right) = 2 a \! \left(n \right)+3 n -1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20527/
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[19,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[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[1,x]+F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]+F[18,x]
F[18,x] = F[15,x]
F[19,x] = F[20,x]+F[35,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[13,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[1,x]+F[30,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[30,x]
F[35,x] = F[36,x]+F[37,x]+F[54,x]
F[36,x] = 0
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[47,x]
F[39,x] = F[40,x]+F[6,x]
F[40,x] = F[36,x]+F[41,x]+F[43,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[18,x]+F[6,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[17,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = x^2
F[47,x] = F[35,x]+F[48,x]
F[48,x] = 2*F[36,x]+F[49,x]+F[69,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[65,x]
F[51,x] = F[36,x]+F[52,x]+F[54,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[51,x]+F[6,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[30,x]+F[51,x]
F[57,x] = F[58,x]+F[62,x]
F[58,x] = F[36,x]+F[59,x]+F[61,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[12,x]+F[33,x]
F[61,x] = F[12,x]*F[29,x]
F[62,x] = 2*F[36,x]+F[63,x]+F[66,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[18,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[33,x]+F[65,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[68,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[12,x]*F[33,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_{19}\! \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_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \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_{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_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{54}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{6}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{6}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = x^{2}
F_{47}\! \left(x \right) = F_{35}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{49}\! \left(x \right)+F_{69}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{65}\! \left(x \right)
F_{51}\! \left(x \right) = F_{36}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \left(x \right)+F_{6}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{30}\! \left(x \right)+F_{51}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{36}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right)+F_{33}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{33}\! \left(x \right)+F_{65}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \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_19(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_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x) + F_18(x))
Eq(F_18(x), F_15(x))
Eq(F_19(x), F_20(x) + F_35(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_13(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_1(x) + F_30(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_30(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_54(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_47(x))
Eq(F_39(x), F_40(x) + F_6(x))
Eq(F_40(x), F_36(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_18(x) + F_6(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_17(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), x**2)
Eq(F_47(x), F_35(x) + F_48(x))
Eq(F_48(x), 2*F_36(x) + F_49(x) + F_69(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_65(x))
Eq(F_51(x), F_36(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_51(x) + F_6(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_30(x) + F_51(x))
Eq(F_57(x), F_58(x) + F_62(x))
Eq(F_58(x), F_36(x) + F_59(x) + F_61(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_12(x) + F_33(x))
Eq(F_61(x), F_12(x)*F_29(x))
Eq(F_62(x), 2*F_36(x) + F_63(x) + F_66(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_18(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_33(x) + F_65(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_68(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_12(x)*F_33(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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