0132_0231_1032_1230_2130
Counting sequence:
1, 1, 2, 6, 19, 59, 181, 551, 1669, 5039, 15181, 45671, 137269, 412319, 1237981, 3715991, 11152069, 33464399, 100409581, 301261511, 903850069, 2711681279, 8135305981, 24406442231, 73220375269, 219663222959, 658993863181, 1976989978151, 5930986711669, 17792993689439, 53379048177181, 160137278749271, 480412104683269, 1441236850920719, 4323711626503981, 12971137026995591, 38913415375954069, 116740254717796799, 350220781333259581, 1050662378359517111, 3151987203798028069, 9455961748833037679, 28367885521377019981, 85103657113886873831, 255310972441172249269, 765932919522540003359, 2297798762965666521181, 6893396297693092585751, 20680188910671463801669, 62040566767198763493839, 186121700371965034659181, 558365101256632592332871, 1675095304051372753709269, 5025285912717068214549119, 15075857739277104550489981, 45227573220083113465155191, 135682719664752940022836069, 407048159003266019323249199, 1221144477027812456479229581, 3663433431119466166456652711, 10990300293430456093407886069, 32970900880435483468299514079, 98912702641594680781050253981, 296738107925360503095454185431, 890214323777234430790969403269, 2670642971334009135382121903759, 8011928914006639092164793099181, 24035786742029140648531234073351, 72107360226105868689667411771669, 216322080678354499557149654418239, 648966242035137285647743801461181, 1946898726105559430895821080796471, 5840696178316973440592642595215269, 17522088534951510617588286491297519, 52566265604855712444385576885195981, 157698796814569498516398165478194791, 473096390443713217915677366079798069, 1419289171331149098479997837529821599, 4257867513993466184905924991170319581, 12773602541980436333649637930672668311, 38320807625941384558812639706341424069, 114962422877824304792165370947671110479, 344887268633473216607951016500307007981, 1034661805900420254286762856815508377031, 3103985417701261971786108185075699837269, 9311956253103788333209963784485448924159, 27935868759311369835333169811973045597181, 83807606277934119177406066352952534440951, 251422818833802376875031312892924398621669, 754268456501407169310720166346906786462639, 2262805369504221585303412954376987540583181, 6788416108512664910652743773803496984140071, 20365248325537995041443241142755559677201269, 61095744976613985743299743070956816481165919, 183287234929841958467839268498250724342621981, 549861704789525877879397884065512722826114391, 1649585114368577638589953809338059268074840069, 4948755343105732925673381742297220003417513999, 14846266029317198796827185855457744408638529581, 44538798087951596430095638823505402022687563911, 133616394263854789369515078984780543661606642069
Generating function in Maple syntax:
(x^4+2*x^3+3*x^2-4*x+1)/(3*x-1)/(2*x-1)
Generating function in latex syntax:
\frac{x^{4}+2 x^{3}+3 x^{2}-4 x +1}{\left(3 x -1\right) \left(2 x -1\right)}
Generating function in sympy syntax:
(x**4 + 2*x**3 + 3*x**2 - 4*x + 1)/((2*x - 1)*(3*x - 1))
Implicit equation for the generating function in Maple syntax:
-(3*x-1)*(2*x-1)*F(x)+x^4+2*x^3+3*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(3 x -1\right) \left(2 x -1\right) F \! \left(x \right)+x^{4}+2 x^{3}+3 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-1/8*2^n+7/27*3^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -\frac{2^{n}}{8}+\frac{7 \,3^{n}}{27} & \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(n+2) = -6*a(n)+5*a(n+1), n >= 5
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(n +2\right) = -6 a \! \left(n \right)+5 a \! \left(n +1\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18866/
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[29,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[18,x]+F[30,x]+F[40,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[20,x]+F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[18,x]+F[30,x]+F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[23,x]+F[33,x]
F[36,x] = 2*F[18,x]+F[37,x]+F[38,x]
F[37,x] = F[12,x]*F[32,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[27,x]+F[36,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[35,x]+F[42,x]
F[42,x] = F[43,x]+F[56,x]
F[43,x] = F[18,x]+F[44,x]+F[54,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[12,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[14,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[23,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[27,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[23,x]+F[43,x]
F[56,x] = F[18,x]+F[57,x]+F[69,x]+F[71,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[64,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[61,x]
F[61,x] = F[10,x]*F[12,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[17,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[33,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[36,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[33,x]+F[56,x]
F[71,x] = F[12,x]*F[42,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_{29}\! \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_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \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_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{20}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{56}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \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_{12}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{33}\! \left(x \right)+F_{56}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \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_29(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_18(x) + F_30(x) + F_40(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_20(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_18(x) + F_30(x) + F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_23(x) + F_33(x))
Eq(F_36(x), 2*F_18(x) + F_37(x) + F_38(x))
Eq(F_37(x), F_12(x)*F_32(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_27(x) + F_36(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_56(x))
Eq(F_43(x), F_18(x) + F_44(x) + F_54(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_12(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_14(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_23(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_27(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_23(x) + F_43(x))
Eq(F_56(x), F_18(x) + F_57(x) + F_69(x) + F_71(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_64(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_10(x)*F_12(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_17(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_33(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_36(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_33(x) + F_56(x))
Eq(F_71(x), F_12(x)*F_42(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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