0132_0231_0321_2013_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 128, 313, 751, 1776, 4155, 9639, 22208, 50877, 116003, 263432, 596175, 1345219, 3027600, 6798833, 15237799, 34093024, 76164531, 169926159, 378663936, 842929589, 1874659707, 4165732088, 9249891239, 20525437131, 45518596240, 100891063497, 223515523679, 494965562704, 1095654027115, 2424487254327, 5363267424192, 11860843581165, 26223483827987, 57964853902824, 128099789032127, 283041537182867, 625284878849808, 1381143447894241, 3050276235296343, 6735732954092480, 14872400565379171, 32834659784654623, 72484217360601600, 159999164960982117, 353149649055418347, 779416834169037272, 1720099470694254615, 3795881865234323483, 8376247114218476048, 18482726798292790329, 40781601660143071375, 89979982831150953264, 198523757253887365787, 437991245754503140807, 966286733513327910592, 2131745742626884538397, 4702779767700954920451, 10374440342300603157192, 22885814573998821664175, 50484785209240059871587, 111364763347863646145936, 245656846443891960447185, 541881488445355673748231, 1195293760935238379606944, 2636576409707695491590163, 5815698390647400200786735, 12828018707803345868896768, 28295270156461001404816405, 62411551365862631361284955, 137661746764115065293197496, 303640014333864045394673031, 669734081331936548957554283, 1477214912024679816822152592, 3258239843576606986266671273, 7186553778871917135946282943, 15851002490542047317625490704, 34961604866207768079339197131, 77112483594381586210267766103, 170081409845493485569447200704, 375135304889571270880805954125, 827404854038277191297024385523, 1824934639251553994813785394088, 4025091626051691513808955586975, 8877762191459684725516093248819, 19580807192806972459048287270416, 43187402352937734458310160885185, 95253959579879349694945676533943, 210091511717654063954558163367808, 463375996518422646578663533679811, 1022017094077078211274746836011583, 2254147982792517623349000019627008, 4971716527944872166966559941405893, 10965539281649651092587659455767499, 24185404809457476903283904405050264, 53342882673591140163053539699282935
Generating function in Maple syntax:
(x^6+5*x^5-3*x^4+x^3-5*x^2+4*x-1)/(x-1)/(2*x-1)/(x^3+2*x-1)
Generating function in latex syntax:
\frac{x^{6}+5 x^{5}-3 x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{3}+2 x -1\right)}
Generating function in sympy syntax:
(x**6 + 5*x**5 - 3*x**4 + x**3 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^3+2*x-1)*F(x)+x^6+5*x^5-3*x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)+x^{6}+5 x^{5}-3 x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
1946/59*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-2020/59*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+3940/59*Sum(_alpha^(1-n),_alpha = RootOf(2*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-6009/59*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+2084/59*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+1/2*piecewise(n = 0,13/2,n = 1,1,0)
Explicit closed form in latex syntax:
\frac{1946 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{59}-\frac{2020 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{59}+\frac{3940 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{59}-\frac{6009 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{59}+\frac{2084 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{59}+\frac{\left(\left\{\begin{array}{cc}\frac{13}{2} & n =0 \\ 1 & n =1 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 128
a(n+4) = -2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)-2, 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) = 51
a \! \left(6\right) = 128
a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18707/
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[17,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[31,x]
F[34,x] = F[35,x]+F[36,x]+F[47,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[25,x]
F[42,x] = F[34,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[31,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[60,x]
F[49,x] = F[18,x]+F[50,x]
F[50,x] = F[35,x]+F[51,x]+F[59,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[11,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[25,x]*F[4,x]
F[56,x] = F[50,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[46,x]
F[59,x] = F[4,x]*F[49,x]
F[60,x] = F[61,x]+F[67,x]
F[61,x] = F[32,x]+F[35,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[53,x]+F[64,x]
F[64,x] = F[31,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[33,x]*F[4,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[50,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_{17}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \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_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \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_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{47}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{25}\! \left(x \right)
F_{42}\! \left(x \right) = F_{34}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \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_{31}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{60}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \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_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \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_{46}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{53}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{31}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{50}\! \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_17(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_34(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(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_33(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_31(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_47(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_25(x))
Eq(F_42(x), F_34(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_31(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_60(x))
Eq(F_49(x), F_18(x) + F_50(x))
Eq(F_50(x), F_35(x) + F_51(x) + F_59(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_11(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_25(x)*F_4(x))
Eq(F_56(x), F_50(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_46(x))
Eq(F_59(x), F_4(x)*F_49(x))
Eq(F_60(x), F_61(x) + F_67(x))
Eq(F_61(x), F_32(x) + F_35(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_53(x) + F_64(x))
Eq(F_64(x), F_31(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_33(x)*F_4(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_50(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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