0231_0312_0321_1203_2130
Counting sequence:
1, 1, 2, 6, 19, 53, 143, 380, 994, 2568, 6571, 16683, 42083, 105582, 263686, 655978, 1626423, 4020825, 9915099, 24395856, 59908478, 146863172, 359480375, 878716695, 2145349671, 5232133402, 12747943978, 31033112270, 75487067707, 183491373565, 445743157511, 1082197273124, 2626060998714, 6369439203136, 15442398854691, 37425079106883, 90669361389771, 219594869978758, 531692079726318, 1287021790510770, 3114652250998943, 7535992451389857, 18230012232620435, 44091683664565368, 106624379216501654, 257805816485911420, 623262427712944623, 1506594502615791407, 3641444468740851663, 8800495927214763314, 21266725128108906194, 51387516829105955734, 124159912564784846307, 299967938320542233541, 724670552575275863807, 1750576723988371090636, 4228619957948438889234, 10213983318047497107832, 24670187630885206711835, 59584356610897878589531, 143905063593003148245683, 347540410314390317397406, 839307735287836035731542, 2026865745762489078597370, 4894604883075153714716551, 11819528675503527804302057, 28541278426136098605600715, 68918883568145843101853856, 166416094806758752278280750, 401832787862379171878640052, 970261897933318862562360615, 2342754087776951397525933127, 5656646796263806482879595255, 13657981353259535780342660170, 32976874352740755378202560506, 79621136481433386031288439390, 192239893833947177846918292235, 464146682035964919872040374669, 1120634180101843713379370401399, 2705637633149925388613662750068, 6532400386108877752719374397546, 15771521206978014114066139897008, 38077830994195845462861992467859, 91932450523338767265923479881331, 221954349498421837666989390679579, 535866772629410297426473476414902, 1293744408303667204399212128906814, 3123480233787142707655729743472450, 7540979788087976477398906849513967, 18206046147929568149709289695222321, 43954409404430455752759810754646403, 106117814509967189464801795467552968, 256196543868684726788764916468549958, 618525250456459770231076982476935916, 1493278690753027231437982473830025695, 3605152429349680053587570629730992735, 8703737492435842106762926486552305055, 21012966946159696767600092701763794530, 50730420246019066463416978787530282914, 122475459103708946105491586824985431846, 295684981316397523997001894632751481267
Generating function in Maple syntax:
-(x^6-4*x^4-2*x^3-x^2+3*x-1)/(x^2+2*x-1)/(x^3+2*x-1)
Generating function in latex syntax:
-\frac{x^{6}-4 x^{4}-2 x^{3}-x^{2}+3 x -1}{\left(x^{2}+2 x -1\right) \left(x^{3}+2 x -1\right)}
Generating function in sympy syntax:
(-x**6 + 4*x**4 + 2*x**3 + x**2 - 3*x + 1)/((x**2 + 2*x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(x^3+2*x-1)*F(x)+x^6-4*x^4-2*x^3-x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)+x^{6}-4 x^{4}-2 x^{3}-x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-1081/236*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3+3*_Z^2-4*_Z+1))-651/59*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3+3*_Z^2-4*_Z+1))-1045/118*Sum(_alpha^(1-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3+3*_Z^2-4*_Z+1))-4191/236*Sum(_alpha^(-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3+3*_Z^2-4*_Z+1))+616/59*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3+3*_Z^2-4*_Z+1))-piecewise(n = 0,-2,n = 1,1,0)
Explicit closed form in latex syntax:
-\frac{1081 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}+3 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{236}-\frac{651 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}+3 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{59}-\frac{1045 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}+3 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{118}-\frac{4191 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}+3 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{236}+\frac{616 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}+3 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{59}-\left(\left\{\begin{array}{cc}-2 & n =0 \\ 1 & n =1 \\ 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) = 53
a(6) = 143
a(n+5) = -a(n)-2*a(n+1)-a(n+2)-3*a(n+3)+4*a(n+4), 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) = 143
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20609/
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[10,x]*F[4,x]
F[4,x] = F[13,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[6,x]
F[13,x] = F[14,x]+F[29,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[16,x]
F[16,x] = F[17,x]+F[22,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[20,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[10,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[14,x]
F[29,x] = F[30,x]+F[31,x]+F[41,x]
F[30,x] = 0
F[31,x] = F[10,x]*F[32,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]*F[36,x]
F[36,x] = F[10,x]
F[37,x] = F[29,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[27,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[44,x]+F[53,x]
F[44,x] = F[28,x]+F[30,x]+F[45,x]
F[45,x] = F[10,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[10,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = x^2
F[50,x] = F[27,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[10,x]*F[27,x]
F[53,x] = F[54,x]
F[54,x] = F[10,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \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_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \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_{10}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)
F_{37}\! \left(x \right) = F_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{27}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{44}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \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_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = x^{2}
F_{50}\! \left(x \right) = F_{27}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_13(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_6(x))
Eq(F_13(x), F_14(x) + F_29(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_22(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_20(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_10(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_14(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_41(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_10(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x)*F_36(x))
Eq(F_36(x), F_10(x))
Eq(F_37(x), F_29(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_27(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_53(x))
Eq(F_44(x), F_28(x) + F_30(x) + F_45(x))
Eq(F_45(x), F_10(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_10(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), x**2)
Eq(F_50(x), F_27(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_10(x)*F_27(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_10(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [1, 2, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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