0123_0231_2031_3012_3102
Counting sequence:
1, 1, 2, 6, 19, 55, 148, 378, 929, 2217, 5170, 11836, 26695, 59475, 131168, 286830, 622717, 1343629, 2883742, 6160560, 13107395, 27787479, 58720492, 123732226, 260047129, 545259825, 1140851018, 2382365028, 4966056319, 10334765467, 21474836920, 44560286166, 92341797365, 191126045205, 395136991798, 816043786840, 1683627180667, 3470333575839, 7146825581252, 14705968022250, 30236569764625, 62122406970169, 127543348822882, 261683767411596, 536561674355639, 1099511627776995, 2251799813686288, 4609152743638078, 9429411719808109, 19281035904681117, 39406496739493070, 80501843339248896, 164381386399024435, 335518172239103335, 684547143360316828, 1396115884484855250, 2846274964498155017, 5800636320053200449, 11817445422220183162, 24067236408667932340, 48999163945790998255, 99727710148492265259, 202914184810805069672, 412745898649251219366, 839326855353784600549, 1706323826818133526565, 3467987885857395705958, 7046656236157048719528, 14314673401198612056299, 29072068660166253349167, 59029581035870565173620, 119830049502817247300026, 243201873867786728507905, 493487297459877924833865, 1001141694368364785306258, 2030617587633947441892060, 4117903573062330626345767, 8349143941713532737817459, 16924961474604808445889472, 34303270131565102832290830, 69513234627841177545608285, 140839857985104298853272749, 285306493429052485230660862, 577866541775792745509555536, 1170240193386961041115581859, 2369494606444673182424108535, 4797017652230848565234110028, 9710092183144701531240009378, 19652298123655411864023600889, 39768823762042841331134369617, 80466102553549717868443078570, 162789115166027506149234839556, 329292050449911153123167047775, 666011741135534587895728836795, 1346878762742493739090247160088, 2723468086427836604778073297270, 5506357294741371462751304552917, 11131556833254139431892925026869, 22500798154051071876566481900182, 45476965283187729778694227497720, 91904668516546631608510982394715
Generating function in Maple syntax:
-(2*x^6-2*x^5+6*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{6}-2 x^{5}+6 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:
(-2*x**6 + 2*x**5 - 6*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)+2*x^6-2*x^5+6*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)+2 x^{6}-2 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,5+3/4*2^n*n-5*2^(-1+n)-1/2*n+1/2*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 5+\frac{3 \,2^{n} n}{4}-5 \,2^{-1+n}-\frac{n}{2}+\frac{n^{2}}{2} & \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) = 55
a(6) = 148
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-5/2*n+6, 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) = 55
a \! \left(6\right) = 148
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{5 n}{2}+6, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17607/
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[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[11,x]
F[22,x] = F[18,x]
F[23,x] = F[24,x]+F[30,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[10,x]+F[27,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]+F[53,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[39,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[11,x]+F[38,x]
F[38,x] = F[36,x]
F[39,x] = F[30,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[52,x]
F[43,x] = F[31,x]+F[44,x]+F[46,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[37,x]+F[42,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[24,x]+F[48,x]
F[48,x] = F[31,x]+F[46,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[21,x]+F[51,x]
F[51,x] = F[28,x]
F[52,x] = F[41,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[24,x]+F[43,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[31,x]+F[49,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[27,x]+F[60,x]
F[60,x] = F[57,x]
F[61,x] = 2*F[31,x]+F[62,x]+F[66,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[38,x]
F[65,x] = F[52,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[43,x]
F[69,x] = F[61,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_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{53}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \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_{52}\! \left(x \right)
F_{43}\! \left(x \right) = F_{31}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{21}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{28}\! \left(x \right)
F_{52}\! \left(x \right) = F_{41}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{24}\! \left(x \right)+F_{43}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \left(x \right)+F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{27}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{38}\! \left(x \right)
F_{65}\! \left(x \right) = F_{52}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{43}\! \left(x \right)
F_{69}\! \left(x \right) = F_{61}\! \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_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_11(x))
Eq(F_22(x), F_18(x))
Eq(F_23(x), F_24(x) + F_30(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_10(x) + F_27(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_53(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_39(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_11(x) + F_38(x))
Eq(F_38(x), F_36(x))
Eq(F_39(x), F_30(x) + F_40(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_52(x))
Eq(F_43(x), F_31(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_37(x) + F_42(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_24(x) + F_48(x))
Eq(F_48(x), F_31(x) + F_46(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_21(x) + F_51(x))
Eq(F_51(x), F_28(x))
Eq(F_52(x), F_41(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_24(x) + F_43(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_31(x) + F_49(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_27(x) + F_60(x))
Eq(F_60(x), F_57(x))
Eq(F_61(x), 2*F_31(x) + F_62(x) + F_66(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_38(x))
Eq(F_65(x), F_52(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_43(x))
Eq(F_69(x), F_61(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, 2, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 2, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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