0132_0231_1230_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 153, 398, 1011, 2551, 6453, 16424, 42078, 108413, 280543, 728289, 1894964, 4938625, 12886020, 33651129, 87932744, 229880476, 601178812, 1572600511, 4114517199, 10766746686, 28177322178, 73748428308, 193034391441, 505287617406, 1322694220083, 3462526580824, 9064348620913, 23729445505093, 62121840370823, 162631780594794, 425764911427764, 1114645773761635, 2918138050054025, 7639699656851259, 20000823481465474, 52362495909548351, 137086114491266022, 358894748052511691, 939595930642891854, 2459888645829518866, 6440061210752495478, 16860277394241761993, 44140735787600525421, 115561859599815444236, 302544702274357242388, 792071965748279344374, 2073670632020527122023, 5428938804413394910948, 14213143529419843635453, 37210487280246508311474, 97418309304120426219839, 255044422617716260501795, 667714922520231335929705, 1748100272885383708938364, 4576585752020731714577754, 11981656694946435282599001, 31368383756357821829277507, 82123493421205525597832421, 215002094201415745749933984, 562882784571355693223950221, 1473646250289279297066467928, 3858055947849738124265185453, 10100521556366446928309223076, 26443508647462626365823467416, 69230004238447479579483906368, 181246503772731907193274513699, 474509506489452431641633016283, 1242282014515033767014212207978, 3252336534694465627966179917558, 8514727584845996634014681185704, 22291846210398791308338572002189, 58360811027460911359522452688730, 152790586834205010907271623006935, 400010949399596257636378091492444, 1047242261213468034550034003135709, 2741715833938576391110066622662801, 7177905239997798228972851275842267, 18791999884845892476193858028415326, 49198094412122027560379464458160464, 128802283346684486926486018645318751, 337208755618260026662161558078128653, 882823983488752779946164588791652063, 2311263194809312686948664074704011406, 6050965600861814028444491368136862571, 15841633607621386893474137495341751898, 41473935221692861642156576049161064799, 108580172056838228013352900514689215818, 284266580947583882358616745220004672558, 744219570783437538983926574595523642138, 1948392131397776974436021457466966716109, 5100956823399989864009854755606180337689, 13354478338782385576964976724953184996440, 34962478192907552785627943250456599220840, 91532956239861044617404588688823065112874, 239636390526517124741557294140825504462339
Generating function in Maple syntax:
-(x^8+2*x^7-23*x^6+57*x^5-81*x^4+69*x^3-34*x^2+9*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
-\frac{x^{8}+2 x^{7}-23 x^{6}+57 x^{5}-81 x^{4}+69 x^{3}-34 x^{2}+9 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**8 - 2*x**7 + 23*x**6 - 57*x**5 + 81*x**4 - 69*x**3 + 34*x**2 - 9*x + 1)/((x - 1)**5*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^5*F(x)+x^8+2*x^7-23*x^6+57*x^5-81*x^4+69*x^3-34*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{8}+2 x^{7}-23 x^{6}+57 x^{5}-81 x^{4}+69 x^{3}-34 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/24*n^4-1/4*n^3-1/24*n^2-2+1/4*n+2^(-1+n)+(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{n^{4}}{24}-\frac{n^{3}}{4}-\frac{n^{2}}{24}-2+\frac{n}{4}+2^{-1+n}+\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 153
a(7) = 398
a(8) = 1011
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1/24*n*(n^3-6*n^2-25*n+54), n >= 9
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) = 56
a \! \left(6\right) = 153
a \! \left(7\right) = 398
a \! \left(8\right) = 1011
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{n \left(n^{3}-6 n^{2}-25 n +54\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19024/
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[38,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[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = F[34,x]
F[38,x] = F[18,x]+F[39,x]+F[69,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[42,x]+F[6,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[43,x]
F[46,x] = F[38,x]+F[47,x]
F[47,x] = 2*F[18,x]+F[48,x]+F[52,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[42,x]
F[51,x] = F[47,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[65,x]
F[54,x] = F[18,x]+F[55,x]+F[63,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[44,x]+F[57,x]
F[57,x] = F[54,x]+F[58,x]
F[58,x] = 2*F[18,x]+F[52,x]+F[59,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[45,x]
F[62,x] = F[58,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[23,x]+F[54,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[61,x]+F[68,x]
F[68,x] = F[65,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[64,x]+F[71,x]
F[71,x] = F[34,x]+F[72,x]
F[72,x] = 2*F[18,x]+F[66,x]+F[73,x]
F[73,x] = F[12,x]*F[71,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_{38}\! \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_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{69}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{6}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{42}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{65}\! \left(x \right)
F_{54}\! \left(x \right) = F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{63}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{44}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{45}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{61}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{65}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{64}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{34}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \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_38(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_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), F_34(x))
Eq(F_38(x), F_18(x) + F_39(x) + F_69(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_42(x) + F_6(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_43(x))
Eq(F_46(x), F_38(x) + F_47(x))
Eq(F_47(x), 2*F_18(x) + F_48(x) + F_52(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_42(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_65(x))
Eq(F_54(x), F_18(x) + F_55(x) + F_63(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_44(x) + F_57(x))
Eq(F_57(x), F_54(x) + F_58(x))
Eq(F_58(x), 2*F_18(x) + F_52(x) + F_59(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_45(x))
Eq(F_62(x), F_58(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_23(x) + F_54(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_61(x) + F_68(x))
Eq(F_68(x), F_65(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_64(x) + F_71(x))
Eq(F_71(x), F_34(x) + F_72(x))
Eq(F_72(x), 2*F_18(x) + F_66(x) + F_73(x))
Eq(F_73(x), F_12(x)*F_71(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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