0132_0213_0231_0321_1320_3120
Counting sequence:
1, 1, 2, 6, 18, 50, 130, 323, 778, 1833, 4248, 9719, 22006, 49397, 110068, 243699, 536562, 1175537, 2564080, 5570543, 12058606, 26017773, 55967724, 120061931, 256901098, 548405225, 1168113640, 2483027943, 5268045798, 11156848613, 23588765668, 49794777059, 104958263266, 220922380257, 464393338848, 974957576159, 2044404432862, 4282082394077, 8959301779420, 18726057410523, 39101382262746, 81570018885593, 170011985444824, 354042744143831, 736672790609878, 1531619697491925, 3181986650783700, 6605865859678163, 13704312928600018, 28411380461731793, 58863454504615888, 121878664915713999, 252201579132747726, 521573131844845517, 1078049160801812428, 2227030015734710219, 4598175219545276362, 9489084414869635017, 19572643980552175560, 40352252661239644103, 83154463519768838086, 171280901028154703813, 352649865221619318724, 725764087150010171331, 1493033348465866833858, 3070229966768033497025, 6311092316217880346560, 12968061083817814785983, 26637098442436592533438, 54694596178548820541373, 112266884432596331134908, 230362939992484880580539, 472531796192143874195386, 968970572703815327285177, 1986345401857044518010808, 4070679908233634174205879, 8339699208747793447387062, 17080799568539506737938357, 34973846172132592452632500, 71591075880303821439631283, 146506697764547873109704626, 299738045400702121003712433, 613076506272068820222869424, 1253656074940370454170304431, 2562922737583013850377093038, 5238275576390188214001860525, 10703829206867926712848482220, 21867050225189412512085311403, 44662555479842860230344966058, 91201363831727624939833917353, 186193919033766726971546402728, 380047592060611744394031136679, 775569434612290742224301326246, 1582396855216537336389805539237, 3227928652436629066799466413988, 6583365128919652301913542623139, 13424221786010663701006103084962, 27368378388521187117469838344097, 55786529930356376708054134030240, 113692413207969324446735568732063, 231663147191708923123522510782366
Generating function in Maple syntax:
-(x^7+10*x^4-23*x^3+19*x^2-7*x+1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
-\frac{x^{7}+10 x^{4}-23 x^{3}+19 x^{2}-7 x +1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(-x**7 - 10*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**2*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)^3*F(x)+x^7+10*x^4-23*x^3+19*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)+x^{7}+10 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/64*(n^2+17*n-4)*2^n-n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(n^{2}+17 n -4\right) 2^{n}}{64}-n +2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 50
a(6) = 130
a(7) = 323
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+n-5, 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) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 130
a \! \left(7\right) = 323
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+n -5, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22160/
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[14,x]*F[4,x]
F[4,x] = F[32,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[14,x]*F[16,x]*F[29,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[15,x]
F[14,x] = x
F[15,x] = F[16,x]+F[18,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[14,x]*F[16,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[28,x]
F[23,x] = 0
F[24,x] = F[14,x]*F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[14,x]*F[16,x]
F[28,x] = F[14,x]*F[18,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]*F[16,x]*F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]*F[14,x]*F[34,x]
F[34,x] = F[35,x]+F[53,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = F[19,x]+F[37,x]
F[37,x] = F[23,x]+F[38,x]+F[43,x]
F[38,x] = F[14,x]*F[39,x]
F[39,x] = F[26,x]+F[40,x]
F[40,x] = F[23,x]+F[38,x]+F[41,x]
F[41,x] = F[14,x]*F[42,x]
F[42,x] = F[19,x]+F[40,x]
F[43,x] = F[14,x]*F[44,x]
F[44,x] = F[42,x]+F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[14,x]*F[48,x]
F[48,x] = F[19,x]+F[46,x]
F[49,x] = 2*F[23,x]+F[50,x]+F[52,x]
F[50,x] = F[14,x]*F[51,x]
F[51,x] = F[40,x]+F[49,x]
F[52,x] = F[14,x]*F[45,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[21,x]*F[26,x]
F[55,x] = F[56,x]
F[56,x] = F[14,x]*F[26,x]*F[57,x]
F[57,x] = F[39,x]+F[58,x]
F[58,x] = F[21,x]^2
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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{32}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{53}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{23}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{23}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{14}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{45}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right) F_{26}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right) F_{26}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{39}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21} \left(x \right)^{2}
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_14(x)*F_4(x))
Eq(F_4(x), F_32(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_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_14(x)*F_16(x)*F_29(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_15(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x) + F_18(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_14(x)*F_16(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_28(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_14(x)*F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_14(x)*F_16(x))
Eq(F_28(x), F_14(x)*F_18(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x)*F_16(x)*F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x)*F_14(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_53(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), F_19(x) + F_37(x))
Eq(F_37(x), F_23(x) + F_38(x) + F_43(x))
Eq(F_38(x), F_14(x)*F_39(x))
Eq(F_39(x), F_26(x) + F_40(x))
Eq(F_40(x), F_23(x) + F_38(x) + F_41(x))
Eq(F_41(x), F_14(x)*F_42(x))
Eq(F_42(x), F_19(x) + F_40(x))
Eq(F_43(x), F_14(x)*F_44(x))
Eq(F_44(x), F_42(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_14(x)*F_48(x))
Eq(F_48(x), F_19(x) + F_46(x))
Eq(F_49(x), 2*F_23(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_14(x)*F_51(x))
Eq(F_51(x), F_40(x) + F_49(x))
Eq(F_52(x), F_14(x)*F_45(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_21(x)*F_26(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_14(x)*F_26(x)*F_57(x))
Eq(F_57(x), F_39(x) + F_58(x))
Eq(F_58(x), F_21(x)**2)
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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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