0132_0213_0231_2031_3012
Counting sequence:
1, 1, 2, 6, 19, 56, 158, 438, 1207, 3319, 9114, 24991, 68417, 186991, 510224, 1389996, 3781093, 10271018, 27863912, 75498924, 204335629, 552438181, 1492066932, 4026096601, 10854136979, 29237722861, 78695343278, 211655833698, 568858426567, 1527870725804, 4101021321794, 11001044672946, 29493399426547, 79027162362211, 211640827197534, 566505529087651, 1515653650100213, 4053178881460603, 10834275484990172, 28948101585388584, 77314998816103609, 206413349485828286, 550869443965172012, 1469611710758154696, 3919271479030902889, 10448703370150920241, 27847034772149346648, 74192488724663220733, 197610498596211130727, 526179120868716516889, 1400667138229806642074, 3727503002675561731326, 9917143722141585296827, 26378153011928456073056, 70144688005836861920294, 186483804233982104769006, 495663031689116298633583, 1317146318583987354792463, 3499342700321700610480482, 9294941083407058447199719, 24684091676718451798035689, 65539108026179284177109191, 173979943513293385356981992, 461759081768691838534706916, 1225325668446281045546060749, 3250944664275657077359372946, 8623660179843708609000830768, 22871764700939018237793541668, 60650668544560976146362290437, 160805615971823250839090397949, 426283269866559168242317614204, 1129870090075726088864291510881, 2994287599208584211408435408507, 7934048607646050049558221120901, 21020065325169671336799968685254, 55681712772087256655245700724186, 147479562102324871312615440126703, 390564875464361382639231523787348, 1034184280967948579991293536992842, 2738087715541578242136661400334954, 7248408893285878497438513197688619, 19186029299101240418436333480619771, 50778019980744299911623030574163622, 134374163238525847609432932943964011, 355554526544289350641923749034017309, 940693454227316339199082081784864851, 2488527892827125963878305818424622388, 6582492356489486308322041572174364128, 17409753516658140931823454172050513697, 46041579081299935643469020567147949734, 121748612050669855496810071124953613684, 321910331153179200185319883971222956736, 851066975332848263575999190683685178337, 2249838302534837576754867826601232282121, 5947006461416099906808325094789240861664, 15718308961457296477817079736252778183941, 41540745533042857088964110143758728143823, 109775275088188643141891866505906021731633, 290066296972988109832840140776817549054002, 766395920104653430492197894222238645959414, 2024757158921140376554332905680116238412899
Generating function in Maple syntax:
(2*x-1)*(x^6-3*x^5+5*x^4-12*x^3+13*x^2-6*x+1)/(x^2-3*x+1)^2/(x-1)^3
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{6}-3 x^{5}+5 x^{4}-12 x^{3}+13 x^{2}-6 x +1\right)}{\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x - 1)*(x**6 - 3*x**5 + 5*x**4 - 12*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*(x-1)^3*F(x)-(2*x-1)*(x^6-3*x^5+5*x^4-12*x^3+13*x^2-6*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-\left(2 x -1\right) \left(x^{6}-3 x^{5}+5 x^{4}-12 x^{3}+13 x^{2}-6 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((-15*n+74)*5^(1/2)+35*n-150)*(3/2-1/2*5^(1/2))^(-n)+1/50*((15*n-74)*5^(1/2)+35*n-150)*(3/2+1/2*5^(1/2))^(-n)+1/2*n^2-7/2*n+5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-15 n +74\right) \sqrt{5}+35 n -150\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(15 n -74\right) \sqrt{5}+35 n -150\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{n^{2}}{2}-\frac{7 n}{2}+5 & \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) = 158
a(7) = 438
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)+1/2*(n+1)*(n-4), 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) = 56
a \! \left(6\right) = 158
a \! \left(7\right) = 438
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)+\frac{\left(n +1\right) \left(n -4\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18196/
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[18,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[14,x]
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[1,x]+F[4,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]+F[33,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[4,x]
F[25,x] = F[19,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[32,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[23,x]
F[31,x] = F[26,x]
F[32,x] = F[25,x]*F[4,x]
F[33,x] = F[27,x]+F[34,x]+F[61,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[37,x]+F[7,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[11,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = 2*F[27,x]+F[42,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[37,x]
F[45,x] = F[41,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[27,x]+F[49,x]+F[59,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[11,x]+F[52,x]
F[52,x] = F[38,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = 2*F[27,x]+F[46,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[52,x]
F[58,x] = F[54,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[19,x]+F[48,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[60,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[27,x]+F[28,x]+F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[74,x]
F[67,x] = F[19,x]+F[68,x]
F[68,x] = F[27,x]+F[69,x]+F[73,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[4,x]
F[72,x] = F[68,x]
F[73,x] = F[19,x]*F[4,x]
F[74,x] = F[64,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_{18}\! \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_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{33}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = 0
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_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{26}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)+F_{61}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \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_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \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_{48}\! \left(x \right) = F_{27}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \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_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{38}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{46}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{65}\! \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_{74}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{27}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right)
F_{72}\! \left(x \right) = F_{68}\! \left(x \right)
F_{73}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{74}\! \left(x \right) = F_{64}\! \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_18(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_14(x))
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_1(x) + F_4(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x) + F_33(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_4(x))
Eq(F_25(x), F_19(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_32(x))
Eq(F_27(x), 0)
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_23(x))
Eq(F_31(x), F_26(x))
Eq(F_32(x), F_25(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_34(x) + F_61(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_37(x) + F_7(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_11(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), 2*F_27(x) + F_42(x) + F_46(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_37(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_27(x) + F_49(x) + F_59(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_11(x) + F_52(x))
Eq(F_52(x), F_38(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), 2*F_27(x) + F_46(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_52(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_19(x) + F_48(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_60(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_27(x) + F_28(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_74(x))
Eq(F_67(x), F_19(x) + F_68(x))
Eq(F_68(x), F_27(x) + F_69(x) + F_73(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_4(x))
Eq(F_72(x), F_68(x))
Eq(F_73(x), F_19(x)*F_4(x))
Eq(F_74(x), F_64(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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