0123_0132_0321_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 48, 99, 206, 447, 966, 2071, 4442, 9545, 20508, 44045, 94598, 203191, 436440, 937425, 2013488, 4324773, 9289182, 19952221, 42855350, 92048969, 197711898, 424665191, 912137948, 1959180223, 4208121318, 9038619717, 19414042568, 41699403315, 89566108176, 192378957379, 413210577038, 887534599967, 1906334711430, 4094614488407, 8794818511226, 18890382199465, 40574633710332, 87150216620493, 187189866240614, 402064933191927, 863594882293656, 1854914614918129, 3984169312697360, 8557593424827269, 18380846665140126, 39480202827492029, 84799489582599542, 182140741902723721, 391219923895414842, 840300897283461831, 1804881486977015228, 3876703205439353727, 8326767076676846022, 17885054974532568709, 38415292333325614472, 82512169359067597907, 177227809000251395568, 380667440051712205859, 817635227411282594894, 1756197916514958402175, 3772135811389235797830, 8102166871830435197079, 17402636411123865194906, 37379105966173971386889, 80286545660252136973788, 172447394003723944942477, 370399591290402190276934, 795580924958102217135671, 1708827501542630542632024, 3670388942749237167180305, 7883624870792600469580016, 16933230257833705346572645, 36370868942168143452912990, 78120954328628154615638301, 167795922470798146868036918, 360408700070222599552226505, 774121499340447047903938650, 1662734821951690393876428583, 3571386520702807089236532380, 7670977663946634924893328191, 16476485527303939497242821542, 35389827375900016436266010309, 76013776094454530355644981576, 163269916373558502725419823731, 350687295938367579872975797200, 753240904779939115679460426243, 1617885413030232778150831844366, 3475054518528478589382728494047, 7464066249368883261363852609030, 16032060699456073218280141441495, 34435247716722318330390575153594, 73963435365003348028314710645609, 158865991497904057907376002394300, 341228109944633072294395998839117, 732923528305444536137462714273318, 1574245739692614738633630714345847
Generating function in Maple syntax:
-(2*x^8+7*x^7+15*x^6+16*x^5+7*x^4+x^3+1)/(x^2+1)/(x^3+2*x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{8}+7 x^{7}+15 x^{6}+16 x^{5}+7 x^{4}+x^{3}+1}{\left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**8 - 7*x**7 - 15*x**6 - 16*x**5 - 7*x**4 - x**3 - 1)/((x**2 + 1)*(x**3 + 2*x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+1)*(x^3+2*x^2+x-1)*F(x)+2*x^8+7*x^7+15*x^6+16*x^5+7*x^4+x^3+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)+2 x^{8}+7 x^{7}+15 x^{6}+16 x^{5}+7 x^{4}+x^{3}+1 = 0
Explicit closed form in Maple syntax:
415/558*Sum(_alpha^(3-n),_alpha = RootOf(_Z^5+2*_Z^4+2*_Z^3+_Z^2+_Z-1))+39/31*Sum(_alpha^(2-n),_alpha = RootOf(_Z^5+2*_Z^4+2*_Z^3+_Z^2+_Z-1))+111/62*Sum(_alpha^(1-n),_alpha = RootOf(_Z^5+2*_Z^4+2*_Z^3+_Z^2+_Z-1))+79/186*Sum(_alpha^(-n),_alpha = RootOf(_Z^5+2*_Z^4+2*_Z^3+_Z^2+_Z-1))-80/279*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5+2*_Z^4+2*_Z^3+_Z^2+_Z-1))-piecewise(n = 0,-2,n = 1,5,n = 2,3,n = 3,2,0)
Explicit closed form in latex syntax:
\frac{415 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{558}+\frac{39 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{31}+\frac{111 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{62}+\frac{79 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{186}-\frac{80 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{279}-\left(\left\{\begin{array}{cc}-2 & n =0 \\ 5 & n =1 \\ 3 & n =2 \\ 2 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 48
a(6) = 99
a(7) = 206
a(8) = 447
a(n+5) = a(n)+2*a(n+1)+2*a(n+2)+a(n+3)+a(n+4), n >= 9
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) = 48
a \! \left(6\right) = 99
a \! \left(7\right) = 206
a \! \left(8\right) = 447
a \! \left(n +5\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16279/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[33,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[15,x]+F[25,x]+F[27,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[16,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[42,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[15,x]+F[37,x]+F[41,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]+F[40,x]
F[40,x] = F[15,x]+F[18,x]+F[25,x]
F[41,x] = F[2,x]*F[4,x]
F[42,x] = F[43,x]+F[50,x]
F[43,x] = F[15,x]+F[44,x]+F[49,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]+F[47,x]
F[47,x] = F[15,x]+F[25,x]+F[48,x]
F[48,x] = F[11,x]*F[4,x]
F[49,x] = F[35,x]*F[4,x]
F[50,x] = F[15,x]+F[51,x]+F[55,x]+F[56,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[14,x]+F[54,x]
F[54,x] = F[32,x]
F[55,x] = 0
F[56,x] = F[4,x]*F[43,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{32}\! \left(x \right)
F_{55}\! \left(x \right) = 0
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_33(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_27(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_16(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_42(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_41(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x) + F_40(x))
Eq(F_40(x), F_15(x) + F_18(x) + F_25(x))
Eq(F_41(x), F_2(x)*F_4(x))
Eq(F_42(x), F_43(x) + F_50(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_49(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x) + F_47(x))
Eq(F_47(x), F_15(x) + F_25(x) + F_48(x))
Eq(F_48(x), F_11(x)*F_4(x))
Eq(F_49(x), F_35(x)*F_4(x))
Eq(F_50(x), F_15(x) + F_51(x) + F_55(x) + F_56(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_14(x) + F_54(x))
Eq(F_54(x), F_32(x))
Eq(F_55(x), 0)
Eq(F_56(x), F_4(x)*F_43(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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