0213_1230_1302_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 59, 179, 527, 1513, 4252, 11742, 31961, 85962, 228905, 604432, 1584630, 4128917, 10701139, 27605977, 70924903, 181559963, 463274348, 1178687004, 2991059041, 7572244404, 19128901249, 48228296702, 121375515414, 304957890295, 765038881499, 1916510097695, 4794756701039, 11980895052973, 29902990547884, 74554640054586, 185694250090841, 462074875761726, 1148789472732233, 2853676532372812, 7083133484907558, 17567948195597753, 43542030351774739, 107846014772726533, 266945072490306535, 660352191412150943, 1632592532194632860, 4034039442788674488, 9962618871164242561, 24591674282995057128, 60672709092946251457, 149622972889099203962, 368817157384753637862, 908740173134179833691, 2238161777163406540859, 5510275160030834042891, 13560999235994268968591, 33362059342320501945073, 82046976480318846135676, 201709515133131489893910, 495734870967429128719001, 1217970488341179805922610, 2991527174419155456026921, 7345518721994875660300552, 18031403714814644125032726, 44250498229260085750601309, 108565583425027059604506259, 266290203660358339906888369, 652996251159562641390996007, 1600891765388227674831971363, 3923848161167425498664117132, 9615333905594800354122327252, 23557075987332140194393150561, 57701351728081456130391138972, 141306071154115606064127417025, 345977943305377265441086680182, 846937148013621688343690988150, 2072867069099191260106300812607, 5072376135993901717333514971739, 12410043870417367969730386491767, 30357017785271294076334568483183, 74245611787175662236710865810613, 181555859960496719559041937596428, 443894101256133061849073756160114, 1085125220169564949977552187801241, 2652243626536831272823098090587814, 6481591800823166446188385562490569, 15837484968284610736238413567299652, 38692656545175220592003723137921734, 94517045414302164777548559626272385, 230851336892896610055328671491148499, 563763145593996620021835314236389085, 1376589070387809384254640925057166119, 3360907597377355698416055852545264423, 8204548329464920945432452630747183164, 20026258695959738244717506828930557296, 48875718665011878937366645743604208641, 119271256352890999824026297991133699536, 291024004968236367568490403605583514753, 710024373811156216747286506761557790962, 1732088741231576253805675943326490196678, 4224938941078156111932686986693781038243
Generating function in Maple syntax:
(x^6+4*x^5-7*x^3-2*x^2+4*x-1)/(x^2+x-1)/(x^2+2*x-1)^2
Generating function in latex syntax:
\frac{x^{6}+4 x^{5}-7 x^{3}-2 x^{2}+4 x -1}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)^{2}}
Generating function in sympy syntax:
(x**6 + 4*x**5 - 7*x**3 - 2*x**2 + 4*x - 1)/((x**2 + x - 1)*(x**2 + 2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^2+2*x-1)^2*F(x)-x^6-4*x^5+7*x^3+2*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)^{2} F \! \left(x \right)-x^{6}-4 x^{5}+7 x^{3}+2 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,(-1/2*5^(1/2)-1/2)^(-n)+(1/2*5^(1/2)-1/2)^(-n)-1/16*(4*2^(1/2)*n+5*2^(1/2)+2*n+16)*(-1-2^(1/2))^(-n)+1/16*(4*2^(1/2)*n+5*2^(1/2)-2*n-16)*(2^(1/2)-1)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-\\\frac{\left(4 \sqrt{2}\, n +5 \sqrt{2}+2 n +16\right) \left(-1-\sqrt{2}\right)^{-n}}{16}+\\\frac{\left(4 \sqrt{2}\, n +5 \sqrt{2}-2 n -16\right) \left(\sqrt{2}-1\right)^{-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) = 59
a(6) = 179
a(n+6) = a(n)+5*a(n+1)+5*a(n+2)-6*a(n+3)-5*a(n+4)+5*a(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) = 59
a \! \left(6\right) = 179
a \! \left(n +6\right) = a \! \left(n \right)+5 a \! \left(n +1\right)+5 a \! \left(n +2\right)-6 a \! \left(n +3\right)-5 a \! \left(n +4\right)+5 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21955/
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[17,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[30,x]+F[8,x]
F[8,x] = F[27,x]+F[9,x]
F[9,x] = F[10,x]*F[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[17,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[17,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = x
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[17,x]*F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[17,x]*F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[20,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[11,x]*F[18,x]
F[29,x] = F[11,x]*F[17,x]*F[19,x]
F[30,x] = F[31,x]
F[31,x] = F[17,x]*F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[13,x]*F[17,x]*F[34,x]*F[37,x]
F[34,x] = F[1,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[16,x]*F[17,x]*F[34,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[16,x]*F[20,x]
F[39,x] = F[17,x]*F[20,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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{30}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{27}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \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_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \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_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right) F_{34}\! \left(x \right) F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \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_17(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_17(x)*F_7(x))
Eq(F_7(x), F_30(x) + F_8(x))
Eq(F_8(x), F_27(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_18(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_17(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_17(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_17(x)*F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_17(x)*F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_20(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_11(x)*F_18(x))
Eq(F_29(x), F_11(x)*F_17(x)*F_19(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_17(x)*F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_13(x)*F_17(x)*F_34(x)*F_37(x))
Eq(F_34(x), F_1(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_16(x)*F_17(x)*F_34(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_16(x)*F_20(x))
Eq(F_39(x), F_17(x)*F_20(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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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": [[{"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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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": []}, "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": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], 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