0132_0231_2031_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 417, 1079, 2718, 6694, 16175, 38457, 90180, 208976, 479325, 1089643, 2457722, 5505162, 12255387, 27132077, 59769024, 131072212, 286261481, 622854399, 1350566166, 2919235886, 6291456327, 13522436449, 28991029628, 62008590744, 132338680245, 281857229267, 599147938290, 1271310320146, 2692944495155, 5695126635093, 12025908429432, 25357486916252, 53395033424577, 112287624987367, 235845244158734, 494780232500022, 1036839464993631, 2170435953230729, 4538783999460276, 9482188277941216, 19791209299969037, 41271268460201019, 85990605385106538, 179018085187978394, 372391394188199115, 774056185954305277, 1607785066971268400, 3337167323881538916, 6922032627268453785, 14348468412802401743, 29723757540645275142, 61537185308390458942, 127325768665018664567, 263298448614588679857, 544178950174431774444, 1124098466991675803432, 2320830988773582964581, 4789235930136842340259, 9878231451471464892386, 20365205457375344986146, 41966342767689229928547, 86441442729402958874789, 177974186823149753993448, 366278550327576856889644, 753512601922887764412785, 1549526502191602335746487, 3185236192695575696640510, 6545199945257328266185286, 13444577376729879923395215, 27606954458855945919269593, 56668397794435742564354852, 116283552274182143742053232, 238536175782711519034215357, 489161609761569329815489547, 1002803967370334900418776154, 2055173893344869597000502442, 4210688629717753415501614331, 8624476797130764532353862989, 17659988372930502984107822496, 36151717709755870840413490676, 73986260160415305492017974857, 151376855428865406740008537759, 309639752326255741259143450358, 633206330094472010610902044494, 1294575796082686422475759161255, 2646096833972500337596878032897, 5407322091598541040759374614620, 11047376910582733573199784579256, 22565171036093911650861236359445, 46081080022358995352845000118643, 94083442985688900892333441028562, 192049065934576754326750535619122, 391941719958065678075261922316947, 799729072418984223669232634696437
Generating function in Maple syntax:
-(4*x^7-5*x^6+13*x^5-34*x^4+42*x^3-26*x^2+8*x-1)/(2*x-1)^3/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{7}-5 x^{6}+13 x^{5}-34 x^{4}+42 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right)^{3} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**7 + 5*x**6 - 13*x**5 + 34*x**4 - 42*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^3*(x-1)^3*F(x)+4*x^7-5*x^6+13*x^5-34*x^4+42*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{3} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{7}-5 x^{6}+13 x^{5}-34 x^{4}+42 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/16*(n^2+n-6)*2^n+1/2*n^2-3/2*n+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n^{2}+n -6\right) 2^{n}}{16}+\frac{n^{2}}{2}-\frac{3 n}{2}+3 & \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) = 156
a(7) = 417
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-1/2*(n-3)*(n-6), 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) = 156
a \! \left(7\right) = 417
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{\left(n -3\right) \left(n -6\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19229/
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[32,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[14,x]
F[31,x] = F[23,x]
F[32,x] = F[18,x]+F[33,x]+F[54,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[53,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[18,x]+F[37,x]+F[43,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[17,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[13,x]+F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[14,x]+F[46,x]
F[49,x] = 2*F[18,x]+F[50,x]+F[52,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[17,x]+F[49,x]
F[52,x] = F[12,x]*F[45,x]
F[53,x] = F[32,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[64,x]
F[56,x] = F[23,x]+F[57,x]
F[57,x] = F[18,x]+F[58,x]+F[63,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[10,x]+F[61,x]
F[61,x] = F[18,x]+F[21,x]+F[37,x]
F[62,x] = F[57,x]
F[63,x] = F[12,x]*F[56,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[23,x]+F[65,x]
F[68,x] = 2*F[18,x]+F[69,x]+F[71,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[57,x]+F[68,x]
F[71,x] = F[12,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_{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_{32}\! \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_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \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_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{54}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \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_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \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_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{56}\! \left(x \right) = F_{23}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \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_{23}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{57}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \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_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_32(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_23(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_54(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_53(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_18(x) + F_37(x) + F_43(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_17(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_13(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_12(x)*F_48(x))
Eq(F_48(x), F_14(x) + F_46(x))
Eq(F_49(x), 2*F_18(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_17(x) + F_49(x))
Eq(F_52(x), F_12(x)*F_45(x))
Eq(F_53(x), F_32(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_64(x))
Eq(F_56(x), F_23(x) + F_57(x))
Eq(F_57(x), F_18(x) + F_58(x) + F_63(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_10(x) + F_61(x))
Eq(F_61(x), F_18(x) + F_21(x) + F_37(x))
Eq(F_62(x), F_57(x))
Eq(F_63(x), F_12(x)*F_56(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_23(x) + F_65(x))
Eq(F_68(x), 2*F_18(x) + F_69(x) + F_71(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_57(x) + F_68(x))
Eq(F_71(x), F_12(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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