0132_1203_1302_1320_2031
Counting sequence:
1, 1, 2, 6, 19, 59, 178, 524, 1514, 4314, 12165, 34033, 94626, 261814, 721520, 1981866, 5428687, 14834807, 40454770, 110119616, 299262614, 812088186, 2200772217, 5956857181, 16105390914, 43498413034, 117369451868, 316405461834, 852245499139, 2293714435619, 6168615887890, 16577836959668, 44522153888066, 119494429228314, 320520751600365, 859239800717641, 2302152810700002, 6164904841712542, 16500657595346120, 44143652196851946, 118041921512141239, 315510326918363471, 842961943023439282, 2251265664036726056, 6010032101204749934, 16038530534535157434, 42785459936132356833, 114097846282052671669, 304168160704773277314, 810606866616133214674, 2159593014516104821940, 5751793601897006384778, 15314711349959422922731, 40765489417894986310139, 108482279691732245776210, 288609767925508249554716, 767631753849605475042842, 2041211260145940966131802, 5426494587384103568130645, 14422824399856996872554401, 38325141708914085288673314, 101817538047158632538665606, 270439121152539546647927072, 718169833961889460670014506, 1906768757090978708532786079, 5061561556680245326568606759, 13433492892393660373709459698, 35646222934851560422101397904, 94571516366546219635828409414, 250860041695145126013833679354, 665317414818684554100369397065, 1764226905456149719548076761613, 4677458603388118806041894455746, 12399300107232880214631552233914, 32863800023455892374157859911180, 87091023674865643379985963563466, 230762683828827023404491918698707, 611358342578220170794551875013587, 1619442875368517002257918775437970, 4289200563129887165922928995252356, 11358719121328351040200329611369714, 30076407443098671395076805174637466, 79628294827239857194085031871995837, 210792401253891745439040725690252281, 557942889960371257919762190398661474, 1476634287488549893638943916681825134, 3907560048060908288517209469625780088, 10339248064494900999875947634257495914, 27354190693260671522923365915956034439, 72362141450977675163255250353181308159, 191404679418868476311885285245900067890, 506230416647448187960712869184703533048, 1338749684289586525285236541072089875486, 3540029457677512781841812801454294707514, 9359908039344826594346024096657651183025, 24745391890705151989527624760921601911141, 65414789973210833685046553009399328574722, 172908847819894825249552635943650366319522, 457002840518925962784309173985182350885220, 1207763065043252742040899175742016059346186, 3191585441497449365349006271098796593706939
Generating function in Maple syntax:
-(x^6+2*x^5-17*x^4+32*x^3-24*x^2+8*x-1)/(2*x-1)/(x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{x^{6}+2 x^{5}-17 x^{4}+32 x^{3}-24 x^{2}+8 x -1}{\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-x**6 - 2*x**5 + 17*x**4 - 32*x**3 + 24*x**2 - 8*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)*(x^2-3*x+1)^2*F(x)+x^6+2*x^5-17*x^4+32*x^3-24*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{6}+2 x^{5}-17 x^{4}+32 x^{3}-24 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((10*n-3)*5^(1/2)-20*n+25)*(3/2-1/2*5^(1/2))^(-n)+1/50*((-10*n+3)*5^(1/2)-20*n+25)*(3/2+1/2*5^(1/2))^(-n)-2^(-1+n)+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(10 n -3\right) \sqrt{5}-20 n +25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(-10 n +3\right) \sqrt{5}-20 n +25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-2^{-1+n}+1 & \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) = 178
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)-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) = 59
a \! \left(6\right) = 178
a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19874/
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[21,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[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[15,x]+F[23,x]+F[27,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[6,x]
F[26,x] = F[22,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[30,x]+F[43,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[33,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = 2*F[15,x]+F[38,x]+F[42,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[34,x]
F[41,x] = F[37,x]
F[42,x] = F[12,x]*F[36,x]
F[43,x] = 2*F[15,x]+F[44,x]+F[48,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[14,x]
F[47,x] = F[43,x]
F[48,x] = F[12,x]*F[29,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_{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_{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_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{43}\! \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_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \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_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_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) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_27(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_6(x))
Eq(F_26(x), F_22(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_43(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_33(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), 2*F_15(x) + F_38(x) + F_42(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_34(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_12(x)*F_36(x))
Eq(F_43(x), 2*F_15(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_14(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_12(x)*F_29(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": [1, 2, 0, 3], "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, 0, 3, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 0, 3, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 0, 3, 1], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 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