0132_0231_1230_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 58, 172, 502, 1452, 4176, 11964, 34184, 97488, 277656, 790064, 2246656, 6385776, 18144800, 51545792, 146408288, 415805120, 1180809984, 3353098176, 9521287296, 27035425024, 76764844416, 217964446464, 618877646848, 1757199267584, 4989248930304, 14166021053440, 40221621245440, 114201167064064, 324250762252288, 920642696985600, 2613972220545024, 7421823913709568, 21072700756563968, 59831468992376832, 169878755130408960, 482334614784733184, 1369486514041004032, 3888365382163808256, 11040185251816497152, 31346253606204391424, 89001006380902449152, 252699386000471736320, 717486034486778953728, 2037148624299219812352, 5784049166435078995968, 16422574307885738819584, 46628397955138567274496, 132391393203356448522240, 375897130171592956051456, 1067279744909123071508480, 3030313247608633172557824, 8603928275644825943670784, 24429019610970216482209792, 69360991854533227008753664, 196935745561570491012481024, 559157054122147083017256960, 1587607218095488010112466944, 4507672147304735066557513728, 12798573825657743144118124544, 36338821150755929835074224128, 103176333597308059906153119744, 292947197442135692590979219456, 831761097690045762446307622912, 2361608268204703143307329404928, 6705283076979794396773464670208, 19038221430554243515051994513408, 54054969950854776712318454595584, 153477560235456869487792122494976, 435766804000107153942357366276096, 1237266914961491282863615295619072, 3512955564318811437902050784968704, 9974288205434217294521326335164416, 28319864394394050725170072082448384, 80408215884495846064330563139928064, 228301982371313394877547574061957120, 648214794736133854566934102176432128, 1840467681228424646613659067384594432, 5225615510712092189396898666250240000, 14837020907407542343176633914402799616, 42126556949239858834165873756554133504, 119609375188755868194389985959455227904, 339605314772883957830528175157108801536, 964236872235001747619441421337120210944, 2737744980226002238357546790705751392256, 7773243061509808564239705495922332925952, 22070466069605306170019972999807747751936, 62664381992834149642341074269487524478976, 177922149815935300502773637415632497541120, 505172003431615834751789841274332837314560, 1434328178448391674220786249354051307175936, 4072468999698976434761611477910688672776192, 11562907291865575663427037738296314733527040, 32830409525563190166623591823292806050349056, 93214946934188430689350067861891740998828032, 264663964218232153031841236763157166032945152, 751456888187301348931374373515984905781116928
Generating function in Maple syntax:
(x^4-4*x^3+8*x^2-5*x+1)/(2*x-1)/(2*x^3-4*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{4}-4 x^{3}+8 x^{2}-5 x +1}{\left(2 x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**4 - 4*x**3 + 8*x**2 - 5*x + 1)/((2*x - 1)*(2*x**3 - 4*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(2*x^3-4*x^2+4*x-1)*F(x)-x^4+4*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-x^{4}+4 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
-26/11*Sum(_alpha^(-n+2),_alpha = RootOf(4*_Z^4-10*_Z^3+12*_Z^2-6*_Z+1))+107/22*Sum(_alpha^(-n+1),_alpha = RootOf(4*_Z^4-10*_Z^3+12*_Z^2-6*_Z+1))-50/11*Sum(_alpha^(-n),_alpha = RootOf(4*_Z^4-10*_Z^3+12*_Z^2-6*_Z+1))+27/22*Sum(_alpha^(-n-1),_alpha = RootOf(4*_Z^4-10*_Z^3+12*_Z^2-6*_Z+1))+piecewise(n = 0,1/4,0)
Explicit closed form in latex syntax:
-\frac{26 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(4 \textit{\_Z}^{4}-10 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{11}+\frac{107 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(4 \textit{\_Z}^{4}-10 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{22}-\frac{50 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(4 \textit{\_Z}^{4}-10 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{11}+\frac{27 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(4 \textit{\_Z}^{4}-10 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{22}+\left(\left\{\begin{array}{cc}\frac{1}{4} & 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(n+4) = -4*a(n)+10*a(n+1)-12*a(n+2)+6*a(n+3), n >= 5
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(n +4\right) = -4 a \! \left(n \right)+10 a \! \left(n +1\right)-12 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19018/
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[12,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[35,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[32,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[14,x]
F[31,x] = F[27,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[23,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[18,x]+F[36,x]+F[63,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[40,x]
F[43,x] = F[35,x]+F[44,x]
F[44,x] = 2*F[18,x]+F[45,x]+F[49,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[39,x]
F[48,x] = F[44,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[62,x]
F[51,x] = F[18,x]+F[52,x]+F[60,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[41,x]+F[54,x]
F[54,x] = F[51,x]+F[55,x]
F[55,x] = 2*F[18,x]+F[49,x]+F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[42,x]
F[59,x] = F[55,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[23,x]+F[51,x]
F[62,x] = F[49,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[61,x]+F[65,x]
F[65,x] = F[34,x]+F[66,x]
F[66,x] = 2*F[18,x]+F[67,x]+F[69,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[51,x]+F[66,x]
F[69,x] = F[12,x]*F[65,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \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_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{63}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{39}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{62}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{42}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{23}\! \left(x \right)+F_{51}\! \left(x \right)
F_{62}\! \left(x \right) = F_{49}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{61}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{34}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{51}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_22(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_35(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_23(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_18(x) + F_36(x) + F_63(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_40(x))
Eq(F_43(x), F_35(x) + F_44(x))
Eq(F_44(x), 2*F_18(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_39(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_62(x))
Eq(F_51(x), F_18(x) + F_52(x) + F_60(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_41(x) + F_54(x))
Eq(F_54(x), F_51(x) + F_55(x))
Eq(F_55(x), 2*F_18(x) + F_49(x) + F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_42(x))
Eq(F_59(x), F_55(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_23(x) + F_51(x))
Eq(F_62(x), F_49(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_61(x) + F_65(x))
Eq(F_65(x), F_34(x) + F_66(x))
Eq(F_66(x), 2*F_18(x) + F_67(x) + F_69(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_51(x) + F_66(x))
Eq(F_69(x), F_12(x)*F_65(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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