0132_0231_0312_1032_1230
Counting sequence:
1, 1, 2, 6, 19, 60, 189, 595, 1873, 5896, 18560, 58425, 183916, 578949, 1822473, 5736961, 18059374, 56849086, 178955183, 563332848, 1773314929, 5582216355, 17572253481, 55315679788, 174128175064, 548137914373, 1725482812088, 5431645680297, 17098272199297, 53823634568385, 169431367355946, 533352837917926, 1678940884172251, 5285136390291172, 16637075746565861, 52371834699562179, 164861247948842305, 518966563443248544, 1633654344626029440, 5142578936123214433, 16188319274058017956, 50959194632488457965, 160414399643902992425, 504968333952204613761, 1589589331509597935734, 5003866723825680894078, 15751667234726134382871, 49584658098937092784984, 156087497415395648885401, 491347682599497451569523, 1546712896250739952406873, 4868896034619863103804324, 15326793132326730009566200, 48247197321698070679258701, 151877305924431955792140528, 478094424864960341140352209, 1504992979008243567864245953, 4737565947362525203828067713, 14913379277290330452859885202, 46945812246081989767819583622, 147780677099819492916526746595, 465198650938322332815508120716, 1464398384699912842280271826701, 4609778263084503863591089691251, 14511116549176599260339051405617, 45679529792152798292863208375416, 143794547784165899254026768976064, 452650718309381801397856502183625, 1424898759677182306122208910181724, 4485437431564749719549292903152085, 14119704165536833761228401384246793, 44447403127130268192387935737712641, 139915937443475321205415831701262270, 440441244558047314005727920976157950, 1386464569028893560599052124538557631, 4364450479884907621584440566927018944, 13738849457010997118546333815068560833, 43248510957418159170031065684031744131, 136141945938524552946069701857848312969, 428561100338854731392731944194366580316, 1349067074497263425902749736887953126488, 4246726942911014062378872816826561077973, 13368267648491120703199974000836751512744, 42081956839728704051566027289755275943545, 132469751356352468159043059973337410313921, 417001402554714881184674132024920564937793, 1312678312989530824313133375469425304752954, 4132178795650331121857554165776276934512358, 13007681646187446895675490668721752388728363, 40946868510788325041442433553027604056129332, 128896607900241290600600816371721435992844597, 405753507714787634173751455496524684191718755, 1277271075669214989934645806423962032844470561, 4020720387432738347817929675577995514795570576, 12656821830438096055641532738536620644840589376, 39842397235123384169046987733836458067820116337, 125419843836611986857081282395313965851554267796, 394809005471515890976825699118283109847597311549, 1242818887611350987505105937021026999903546559145, 3912268377867468263947027933124572635922948569729, 12315425854107335984249619620553492653828751913030
Generating function in Maple syntax:
-(x^2-3*x+1)/(x^3-3*x^2+4*x-1)
Generating function in latex syntax:
-\frac{x^{2}-3 x +1}{x^{3}-3 x^{2}+4 x -1}
Generating function in sympy syntax:
(-x**2 + 3*x - 1)/(x**3 - 3*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^3-3*x^2+4*x-1)*F(x)+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
1/31*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^3-3*_Z^2+4*_Z-1))+12/31*Sum(_alpha^(-n),_alpha = RootOf(_Z^3-3*_Z^2+4*_Z-1))-2/31*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3-3*_Z^2+4*_Z-1))
Explicit closed form in latex syntax:
\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{31}+\frac{12 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{31}-\frac{2 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{31}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+3) = a(n)-3*a(n+1)+4*a(n+2), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +3\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 3
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18591/
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[20,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[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[17,x]
F[20,x] = F[21,x]+F[38,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[16,x]+F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[37,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[14,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[26,x]+F[27,x]+F[34,x]
F[34,x] = F[12,x]*F[21,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[25,x]
F[37,x] = F[12,x]*F[24,x]
F[38,x] = F[26,x]+F[39,x]+F[66,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[18,x]+F[26,x]+F[43,x]
F[43,x] = F[12,x]*F[13,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[26,x]+F[39,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[21,x]+F[45,x]
F[48,x] = F[26,x]+F[49,x]+F[62,x]+F[64,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[56,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[53,x]
F[53,x] = F[10,x]*F[12,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[17,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = 2*F[26,x]+F[49,x]+F[58,x]
F[58,x] = F[12,x]*F[45,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = 2*F[26,x]+F[49,x]+F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[45,x]+F[61,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[25,x]+F[61,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[47,x]+F[65,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_{20}\! \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_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \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_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)+F_{39}\! \left(x \right)+F_{66}\! \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_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{26}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{26}\! \left(x \right)+F_{49}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{49}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{45}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{25}\! \left(x \right)+F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{47}\! \left(x \right)+F_{65}\! \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_20(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))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_38(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_16(x) + F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_37(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_14(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_26(x) + F_27(x) + F_34(x))
Eq(F_34(x), F_12(x)*F_21(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_25(x))
Eq(F_37(x), F_12(x)*F_24(x))
Eq(F_38(x), F_26(x) + F_39(x) + F_66(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_18(x) + F_26(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_13(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_26(x) + F_39(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_21(x) + F_45(x))
Eq(F_48(x), F_26(x) + F_49(x) + F_62(x) + F_64(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_56(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_10(x)*F_12(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_17(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), 2*F_26(x) + F_49(x) + F_58(x))
Eq(F_58(x), F_12(x)*F_45(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), 2*F_26(x) + F_49(x) + F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_45(x) + F_61(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_25(x) + F_61(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_47(x) + F_65(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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