0213_0231_0321_1320_3021_3120
Counting sequence:
1, 1, 2, 6, 18, 53, 150, 407, 1064, 2697, 6666, 16139, 38412, 90125, 208910, 479247, 1089552, 2457617, 5505042, 12255251, 27131924, 59768853, 131072022, 286261271, 622854168, 1350565913, 2919235610, 6291456027, 13522436124, 28991029277, 62008590366, 132338679839, 281857228832, 599147937825, 1271310319650, 2692944494627, 5695126634532, 12025908428837, 25357486915622, 53395033423911, 112287624986664, 235845244157993, 494780232499242, 1036839464992811, 2170435953229868, 4538783999459373, 9482188277940270, 19791209299968047, 41271268460199984, 85990605385105457, 179018085187977266, 372391394188197939, 774056185954304052, 1607785066971267125, 3337167323881537590, 6922032627268452407, 14348468412802400312, 29723757540645273657, 61537185308390457402, 127325768665018662971, 263298448614588678204, 544178950174431772733, 1124098466991675801662, 2320830988773582962751, 4789235930136842338368, 9878231451471464890433, 20365205457375344984130, 41966342767689229926467, 86441442729402958872644, 177974186823149753991237, 366278550327576856887366, 753512601922887764410439, 1549526502191602335744072, 3185236192695575696638025, 6545199945257328266182730, 13444577376729879923392587, 27606954458855945919266892, 56668397794435742564352077, 116283552274182143742050382, 238536175782711519034212431, 489161609761569329815486544, 1002803967370334900418773073, 2055173893344869597000499282, 4210688629717753415501611091, 8624476797130764532353859668, 17659988372930502984107819093, 36151717709755870840413487190, 73986260160415305492017971287, 151376855428865406740008534104, 309639752326255741259143446617, 633206330094472010610902040666, 1294575796082686422475759157339, 2646096833972500337596878028892, 5407322091598541040759374610525, 11047376910582733573199784575070, 22565171036093911650861236355167, 46081080022358995352845000114272, 94083442985688900892333441024097, 192049065934576754326750535614562, 391941719958065678075261922312291, 799729072418984223669232634691684
Generating function in Maple syntax:
(4*x^6-3*x^5-10*x^4+23*x^3-19*x^2+7*x-1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
\frac{4 x^{6}-3 x^{5}-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:
(4*x**6 - 3*x**5 - 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)+4*x^6-3*x^5-10*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)+4 x^{6}-3 x^{5}-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),1,1/16*(n^2+n-6)*2^n+n)
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}+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) = 18
a(5) = 53
a(6) = 150
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-n+3, n >= 7
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) = 53
a \! \left(6\right) = 150
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-n +3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22508/
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[23,x] = F[12,x]*F[24,x]*F[9,x]
F[24,x] = F[25,x]+F[38,x]
F[25,x] = F[26,x]+F[36,x]
F[26,x] = F[27,x]*F[5,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[9,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[18,x]+F[33,x]+F[35,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[10,x]+F[32,x]
F[35,x] = F[12,x]*F[31,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[28,x]*F[6,x]*F[9,x]
F[38,x] = F[27,x]*F[39,x]
F[39,x] = F[40,x]+F[6,x]
F[40,x] = F[18,x]+F[41,x]+F[51,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[10,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[43,x]
F[46,x] = F[17,x]+F[47,x]
F[47,x] = 2*F[18,x]+F[48,x]+F[50,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[44,x]+F[47,x]
F[50,x] = F[12,x]*F[46,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[13,x]+F[53,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[14,x]+F[54,x]
F[57,x] = 2*F[18,x]+F[58,x]+F[60,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[17,x]+F[57,x]
F[60,x] = F[12,x]*F[53,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_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{6}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{51}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{17}\! \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_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{17}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \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))
Eq(F_23(x), F_12(x)*F_24(x)*F_9(x))
Eq(F_24(x), F_25(x) + F_38(x))
Eq(F_25(x), F_26(x) + F_36(x))
Eq(F_26(x), F_27(x)*F_5(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_9(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_10(x) + F_32(x))
Eq(F_35(x), F_12(x)*F_31(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_28(x)*F_6(x)*F_9(x))
Eq(F_38(x), F_27(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_6(x))
Eq(F_40(x), F_18(x) + F_41(x) + F_51(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_10(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_43(x))
Eq(F_46(x), F_17(x) + F_47(x))
Eq(F_47(x), 2*F_18(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_44(x) + F_47(x))
Eq(F_50(x), F_12(x)*F_46(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_13(x) + F_53(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_14(x) + F_54(x))
Eq(F_57(x), 2*F_18(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_17(x) + F_57(x))
Eq(F_60(x), F_12(x)*F_53(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": 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:
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