0213_0231_1032_1230_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 131, 299, 649, 1365, 2815, 5735, 11597, 23345, 46867, 93939, 188113, 376493, 753287, 1506911, 3014197, 6028809, 12058075, 24116651, 48233849, 96468293, 192937231, 385875159, 771751069, 1543502945, 3087006755, 6174014435, 12348029857, 24696060765, 49392122647, 98784246479, 197568494213, 395136989753, 790273980907, 1580547963291, 3161095928137, 6322191857909, 12644383717535, 25288767436871, 50577534875629, 101155069753233, 202310139508531, 404620279019219, 809240558040689, 1618481116083725, 3236962232169895, 6473924464342335, 12947848928687317, 25895697857377385, 51791395714757627, 103582791429518219, 207165582859039513, 414331165718082213, 828662331436167727, 1657324662872338871, 3314649325744681277, 6629298651489366209, 13258597302978736195, 26517194605957476291, 53034389211914956609, 106068778423829917373, 212137556847659839031, 424275113695319682479, 848550227390639369509, 1697100454781278743705, 3394200909562557492235, 6788401819125114989435, 13576803638250229983977, 27153607276500459973205, 54307214553000919951807, 108614429106001839909159, 217228858212003679824013, 434457716424007359653873, 868915432848014719313747, 1737830865696029438633651, 3475661731392058877273617, 6951323462784117754553709, 13902646925568235509114055, 27805293851136471018234911, 55610587702272942036476789, 111221175404545884072960713, 222442350809091768145928731, 444884701618183536291864939, 889769403236367072583737529, 1779538806472734145167482885, 3559077612945468290334973775, 7118155225890936580669955735, 14236310451781873161339919837, 28472620903563746322679848225, 56945241807127492645359705187, 113890483614254985290719419299, 227780967228509970581438847713, 455561934457019941162877704733, 911123868914039882325755418967, 1822247737828079764651510847631, 3644495475656159529303021705157
Generating function in Maple syntax:
-(x^6-2*x^4+2*x^3-6*x^2+4*x-1)/(x-1)^3/(2*x-1)
Generating function in latex syntax:
-\frac{x^{6}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1}{\left(x -1\right)^{3} \left(2 x -1\right)}
Generating function in sympy syntax:
(-x**6 + 2*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)^3*(2*x-1)*F(x)+x^6-2*x^4+2*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{3} \left(2 x -1\right) F \! \left(x \right)+x^{6}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1+23/8*2^n-n^2-3*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 1+\frac{23 \,2^{n}}{8}-n^{2}-3 n & \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) = 53
a(6) = 131
a(n+1) = n^2+2*a(n)+n-5, 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) = 53
a \! \left(6\right) = 131
a \! \left(n +1\right) = n^{2}+2 a \! \left(n \right)+n -5, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21846/
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[9,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[10,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]+F[5,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[17,x]*F[9,x]
F[13,x] = F[14,x]+F[19,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[17,x]*F[9,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[17,x]*F[9,x]
F[19,x] = F[20,x]*F[9,x]
F[20,x] = 2*F[19,x]+F[1,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[9,x]
F[23,x] = F[24,x]+F[37,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]+F[33,x]
F[26,x] = F[14,x]*F[27,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]^2*F[9,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[9,x]
F[33,x] = F[34,x]
F[34,x] = F[17,x]*F[31,x]*F[35,x]*F[9,x]
F[35,x] = F[15,x]
F[36,x] = F[30,x]*F[7,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[9,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[30,x]^2*F[17,x]*F[42,x]
F[42,x] = F[1,x]+F[19,x]
F[43,x] = F[17,x]*F[35,x]*F[44,x]
F[44,x] = F[30,x]+F[31,x]
F[45,x] = F[30,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]+F[7,x]
F[48,x] = F[35,x]*F[49,x]
F[49,x] = F[18,x]
F[50,x] = F[49,x]*F[51,x]
F[51,x] = F[19,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_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{5}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{1}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30} \left(x \right)^{2} F_{9}\! \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_{30}\! \left(x \right) F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{17}\! \left(x \right) F_{31}\! \left(x \right) F_{35}\! \left(x \right) F_{9}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right) F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{9}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30} \left(x \right)^{2} F_{17}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{43}\! \left(x \right) = F_{17}\! \left(x \right) F_{35}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{45}\! \left(x \right) = F_{30}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{35}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)
F_{50}\! \left(x \right) = F_{49}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \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_9(x))
Eq(F_4(x), F_21(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_8(x))
Eq(F_8(x), F_10(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x) + F_5(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_17(x)*F_9(x))
Eq(F_13(x), F_14(x) + F_19(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_17(x)*F_9(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_17(x)*F_9(x))
Eq(F_19(x), F_20(x)*F_9(x))
Eq(F_20(x), F_1(x) + 2*F_19(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_9(x))
Eq(F_23(x), F_24(x) + F_37(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x) + F_33(x))
Eq(F_26(x), F_14(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)**2*F_9(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_9(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_17(x)*F_31(x)*F_35(x)*F_9(x))
Eq(F_35(x), F_15(x))
Eq(F_36(x), F_30(x)*F_7(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_9(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_17(x)*F_30(x)**2*F_42(x))
Eq(F_42(x), F_1(x) + F_19(x))
Eq(F_43(x), F_17(x)*F_35(x)*F_44(x))
Eq(F_44(x), F_30(x) + F_31(x))
Eq(F_45(x), F_30(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x) + F_7(x))
Eq(F_48(x), F_35(x)*F_49(x))
Eq(F_49(x), F_18(x))
Eq(F_50(x), F_49(x)*F_51(x))
Eq(F_51(x), F_19(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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 2, 3, 1], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 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