0123_0321_1302_3021_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 266, 565, 1163, 2342, 4638, 9067, 17547, 33684, 64238, 121849, 230099, 432902, 811902, 1518679, 2834319, 5279520, 9817982, 18231913, 33814979, 62650538, 115968666, 214491091, 396438447, 732283272, 1351926038, 2494746061, 4601766911, 8485348862, 15641583234, 28825330267, 53108615031, 97828512468, 180171794378, 331771242433, 610843206455, 1124520221006, 2069940304818, 3809843344951, 7011649118379, 12903317628432, 23744040199658, 43690121914657, 80387824818839, 147903446977442, 272113198831326, 500617735792291, 920979451886139, 1694268721959528, 3116769315372818, 5733479230403125, 10546882414654979, 19400957848535078, 35687511528616854, 65645370714934747, 120750051050238195, 222109163175069156, 408547025779672982, 751474910725690585, 1382242211240573867, 2542443930026788838, 4676451945834045846, 8601608763223252519, 15821266209046923735, 29100559164188902608, 53525427952506595910, 98450479387874619817, 181081686382749832955, 333066039663442960874, 612611871252559040466, 1126781709057555373315, 2072495397550852540503, 3811946867197065659160, 7011317640718867443710, 12895911461716278218989, 23719421192795097768215, 43627047074642622452894, 80243021731729263908442, 147590528781154628619883, 271462278372089424939903, 499298548451523871917252, 918355755955881389884754, 1689123702697158705598657, 3106789527373341450665135, 5714287626212823048269774, 10510231016738542189919274, 19331356970966367176361127, 35555954575014611887442835, 65397670324458061214122848, 120285188593274459711219090, 221239147977321092206476673, 406922548102462991958802799, 748447760365041882097174658, 1376610873344218683310114422, 2531983474403099612634428275
Generating function in Maple syntax:
(2*x^7+2*x^6-4*x^5-6*x^4-3*x^3+2*x-1)/(x-1)/(x^2+x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{2 x^{7}+2 x^{6}-4 x^{5}-6 x^{4}-3 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(2*x**7 + 2*x**6 - 4*x**5 - 6*x**4 - 3*x**3 + 2*x - 1)/((x - 1)*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2+x-1)*(x^3+x^2+x-1)*F(x)+2*x^7+2*x^6-4*x^5-6*x^4-3*x^3+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{7}+2 x^{6}-4 x^{5}-6 x^{4}-3 x^{3}+2 x -1 = 0
Explicit closed form in Maple syntax:
3363/55*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+3*_Z-1))+10803/110*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+3*_Z-1))-14/11*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+3*_Z-1))-6736/55*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+3*_Z-1))-2953/22*Sum(_alpha^(-n),_alpha = RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+3*_Z-1))+5644/55*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+3*_Z-1))+piecewise(n = 0,2,0)
Explicit closed form in latex syntax:
\frac{3363 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{55}+\frac{10803 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{110}-\frac{14 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{11}-\frac{6736 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{55}-\frac{2953 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{22}+\frac{5644 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{55}+\left(\left\{\begin{array}{cc}2 & n =0 \\ 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) = 51
a(6) = 120
a(7) = 266
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)+8, n >= 8
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) = 51
a \! \left(6\right) = 120
a \! \left(7\right) = 266
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+8, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17848/
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[26,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[24,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[4,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[15,x]
F[26,x] = F[27,x]+F[37,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[1,x]+F[15,x]
F[31,x] = F[27,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[27,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[27,x]*F[4,x]
F[37,x] = F[18,x]+F[38,x]+F[63,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[41,x]+F[7,x]
F[41,x] = F[18,x]+F[19,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[14,x]
F[44,x] = F[37,x]+F[45,x]
F[45,x] = 2*F[18,x]+F[46,x]+F[60,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[57,x]
F[48,x] = F[18,x]+F[49,x]+F[56,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[11,x]+F[17,x]
F[52,x] = F[48,x]+F[53,x]
F[53,x] = 2*F[18,x]+F[46,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[32,x]
F[56,x] = F[34,x]*F[4,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[35,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[32,x]+F[53,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[62,x]+F[65,x]
F[65,x] = F[27,x]+F[48,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_{26}\! \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_{14}\! \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_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{63}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{7}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{60}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{57}\! \left(x \right)
F_{48}\! \left(x \right) = F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{32}\! \left(x \right)
F_{56}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{35}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{32}\! \left(x \right)+F_{53}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{27}\! \left(x \right)+F_{48}\! \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_26(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_14(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_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_24(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_27(x) + F_37(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_1(x) + F_15(x))
Eq(F_31(x), F_27(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_27(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_27(x)*F_4(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_63(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_41(x) + F_7(x))
Eq(F_41(x), F_18(x) + F_19(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_14(x))
Eq(F_44(x), F_37(x) + F_45(x))
Eq(F_45(x), 2*F_18(x) + F_46(x) + F_60(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_57(x))
Eq(F_48(x), F_18(x) + F_49(x) + F_56(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_11(x) + F_17(x))
Eq(F_52(x), F_48(x) + F_53(x))
Eq(F_53(x), 2*F_18(x) + F_46(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_32(x))
Eq(F_56(x), F_34(x)*F_4(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_35(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_32(x) + F_53(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_62(x) + F_65(x))
Eq(F_65(x), F_27(x) + F_48(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:
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