0123_0231_0321_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 49, 98, 201, 435, 930, 1963, 4165, 8858, 18809, 39923, 84778, 180035, 382277, 811714, 1723617, 3659955, 7771554, 16502171, 35040869, 74406058, 157994377, 335486515, 712374746, 1512662131, 3211998661, 6820383314, 14482455729, 30752160627, 65299380306, 138657219019, 294425832037, 625186132922, 1327525163609, 2818877398643, 5985626492170, 12709926483875, 26988358100549, 57307292366050, 121686756418689, 258390617953587, 548668675305474, 1165047390828539, 2473870814878565, 5253036792225226, 11154339739371817, 23685212943011251, 50293367896590650, 106793333902795987, 226765807164152773, 481516302746152562, 1022455513509078609, 2171090098389128691, 4610109831718817970, 9789143562618409003, 20786344618133641573, 44137888040967943898, 93722739447776039417, 199011603850248084467, 422582808616227294826, 897315667443963345923, 1905367162656238888517, 4045871655033152993026, 8591035769809773163041, 18242273085043330624947, 38735786490231887551074, 82251874424961070488155, 174654278624899165615205, 370862272200410672455402, 787491872655695754149065, 1672166450955826898811187, 3550691425262142329101466, 7539566165930182081121587, 16009574238415369716143557, 33994856156781490253090450, 72184945577581508789680113, 153278082543054102637300851, 325471888911029331365448978, 691109574922809432088290955, 1467507520075109986307701861, 3116117037906086286401492474, 6616787485648873799616674201, 14050138713550798922810152307, 29834175315162065267845215178, 63350123075822784362691700835, 134518150789384195075390569029, 285636901922632707819393004834, 606523649493986602155857405889, 1287897099147033268851414950451, 2734730854065053676796839504066, 5806949056045325565598755275963, 12330521407392418984410139492709, 26182726369861918441305403677706, 55596607597475095910506732730473, 118054274894212503451829925945523, 250677378046196274213767745016826, 532290320875082806369221943778707
Generating function in Maple syntax:
-(4*x^7+14*x^6+18*x^5+7*x^4+x^3+1)/(2*x^4+2*x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{4 x^{7}+14 x^{6}+18 x^{5}+7 x^{4}+x^{3}+1}{2 x^{4}+2 x^{3}+x^{2}+x -1}
Generating function in sympy syntax:
(-4*x**7 - 14*x**6 - 18*x**5 - 7*x**4 - x**3 - 1)/(2*x**4 + 2*x**3 + x**2 + x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4+2*x^3+x^2+x-1)*F(x)+4*x^7+14*x^6+18*x^5+7*x^4+x^3+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{7}+14 x^{6}+18 x^{5}+7 x^{4}+x^{3}+1 = 0
Explicit closed form in Maple syntax:
154/159*Sum(_alpha^(2-n),_alpha = RootOf(2*_Z^4+2*_Z^3+_Z^2+_Z-1))+349/318*Sum(_alpha^(1-n),_alpha = RootOf(2*_Z^4+2*_Z^3+_Z^2+_Z-1))-95/318*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4+2*_Z^3+_Z^2+_Z-1))+31/106*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4+2*_Z^3+_Z^2+_Z-1))-piecewise(n = 0,-3,n = 1,3,n = 2,5,n = 3,2,0)
Explicit closed form in latex syntax:
\frac{154 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(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)}{159}+\frac{349 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(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)}{318}-\frac{95 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(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)}{318}+\frac{31 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(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)}{106}-\left(\left\{\begin{array}{cc}-3 & n =0 \\ 3 & n =1 \\ 5 & 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) = 49
a(6) = 98
a(7) = 201
a(n+4) = 2*a(n)+2*a(n+1)+a(n+2)+a(n+3), 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) = 49
a \! \left(6\right) = 98
a \! \left(7\right) = 201
a \! \left(n +4\right) = 2 a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17161/
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[20,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[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[38,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[22,x]+F[27,x]+F[33,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[4,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[14,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[4,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[63,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[22,x]+F[42,x]+F[49,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[22,x]+F[27,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[15,x]+F[48,x]
F[48,x] = F[36,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[2,x]+F[51,x]
F[51,x] = F[22,x]+F[52,x]+F[62,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]+F[55,x]
F[55,x] = F[22,x]+F[56,x]+F[61,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = x^2
F[61,x] = F[15,x]*F[4,x]
F[62,x] = F[2,x]*F[4,x]
F[63,x] = F[64,x]+F[71,x]
F[64,x] = F[22,x]+F[65,x]+F[70,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[15,x]+F[68,x]
F[68,x] = F[22,x]+F[56,x]+F[69,x]
F[69,x] = F[4,x]*F[47,x]
F[70,x] = F[4,x]*F[50,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[17,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_{20}\! \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_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{38}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{33}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \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_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{4}\! \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_{63}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{36}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{22}\! \left(x \right)+F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \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_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \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_{60}\! \left(x \right)
F_{60}\! \left(x \right) = x^{2}
F_{61}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{71}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)+F_{65}\! \left(x \right)+F_{70}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \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_20(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))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_11(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_38(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_33(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_4(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_14(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_4(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_63(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_22(x) + F_42(x) + F_49(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_22(x) + F_27(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_15(x) + F_48(x))
Eq(F_48(x), F_36(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_2(x) + F_51(x))
Eq(F_51(x), F_22(x) + F_52(x) + F_62(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x) + F_55(x))
Eq(F_55(x), F_22(x) + F_56(x) + F_61(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), x**2)
Eq(F_61(x), F_15(x)*F_4(x))
Eq(F_62(x), F_2(x)*F_4(x))
Eq(F_63(x), F_64(x) + F_71(x))
Eq(F_64(x), F_22(x) + F_65(x) + F_70(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_15(x) + F_68(x))
Eq(F_68(x), F_22(x) + F_56(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_47(x))
Eq(F_70(x), F_4(x)*F_50(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_17(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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