0132_0231_1023_1302_3012
Counting sequence:
1, 1, 2, 6, 19, 54, 147, 396, 1060, 2824, 7495, 19831, 52343, 137894, 362738, 953126, 2502261, 6564909, 17214981, 45125076, 118250344, 309806164, 811528579, 2125500451, 5566414547, 14576626754, 38169232862, 99942606146, 261681654225, 685148493849, 1793856101985, 4696604361456, 12296326081108, 32193112079344, 84284486551903, 220663300366351, 577711320127151, 1512482471175134, 3959759715718346, 10366843920620126, 27140866535422509, 71055944664208389, 186027345414324669, 487026847493009676, 1275054708893192512, 3338140302843544204, 8739372246951392827, 22879988532638539771, 59900617540220037515, 156821912466533194874, 410565216616402791350, 1074873930896721667706, 2814056963101855188873, 7367297732465029853169, 19287837782405606279193, 50496218710976532801576, 132200824542973479759916, 346106267302842881746984, 906118002135353116018711, 2372247788642812367384551, 6210625462872275788285799, 16259628798132398601774614, 42568261327841687225641634, 111445155978026197492357526, 291767208191503974085845477, 763856471767019862434008029, 1999802213450623888553836917, 5235550181266988353902819396, 13706848355714614274505254680, 35884994936605400672314211524, 93948136555558680147839913715, 245959414932984824582010597331, 643930108649224163219802013763, 1685830911826344404320615714994, 4413562628453122528228485673358, 11554856976779650137337722389426, 30251008308379081797730443663681, 79198167961344103083745132939209, 207343495601626243109288003829201, 542832318895480657555684975898976, 1421153461188707792180899118568964, 3720628064878426844233276769210464, 9740730733862140991011459967867599, 25501564137539132629786160692002751, 66763961680417529900317137223361567, 174790320907038003075105481208523854, 457607001047345571332879766863093882, 1198030682248296894939294740302525646, 3136485045724141481516526295888018845, 8211424454977320285673327831048602549, 21497788319314204847629544564631932205, 56281940503178065201467480597595480956, 147348033190645532645277246697651084528, 385762159069609616511372958434350920444, 1009938444019885484442859026483387972523, 2644053172993451171925238916771785588651, 6922221074967276701548927315343913976571, 18122610051921996273153682212283846707434, 47445609080825946798776397687555406878566, 124214217190610313484904067582477935394026, 325197042491113932379392918524069522235129
Generating function in Maple syntax:
-(x^9-3*x^8+2*x^7+5*x^6-4*x^5+3*x^4-11*x^3+13*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{9}-3 x^{8}+2 x^{7}+5 x^{6}-4 x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**9 + 3*x**8 - 2*x**7 - 5*x**6 + 4*x**5 - 3*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)+x^9-3*x^8+2*x^7+5*x^6-4*x^5+3*x^4-11*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}-3 x^{8}+2 x^{7}+5 x^{6}-4 x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
-1/2*piecewise(n = 0,821/16,n = 1,133/8,n = 2,21/4,n = 3,5/2,n = 4,1,0)+n-4+67/10*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)-67/10*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)+31/2*(3/2+1/2*5^(1/2))^(-n)-11/32*2^n+31/2*(3/2-1/2*5^(1/2))^(-n)
Explicit closed form in latex syntax:
-\frac{\left(\left\{\begin{array}{cc}\frac{821}{16} & n =0 \\ \frac{133}{8} & n =1 \\ \frac{21}{4} & n =2 \\ \frac{5}{2} & n =3 \\ 1 & n =4 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{2}+n -4+\frac{67 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{10}-\frac{67 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{10}+\frac{31 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{2}-\frac{11 \,2^{n}}{32}+\frac{31 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 147
a(7) = 396
a(8) = 1060
a(9) = 2824
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-4+n, n >= 10
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) = 54
a \! \left(6\right) = 147
a \! \left(7\right) = 396
a \! \left(8\right) = 1060
a \! \left(9\right) = 2824
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-4+n, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18785/
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[29,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[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[12,x]+F[21,x]
F[21,x] = F[18,x]+F[19,x]+F[22,x]
F[22,x] = F[12,x]*F[14,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[14,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[12,x]+F[28,x]
F[28,x] = F[26,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[18,x]+F[31,x]+F[63,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[44,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[12,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[37,x]
F[40,x] = F[28,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[28,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[18,x]+F[31,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[2,x]+F[45,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[59,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[12,x]+F[56,x]
F[56,x] = F[35,x]
F[57,x] = F[52,x]+F[58,x]
F[58,x] = F[49,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[52,x]+F[60,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = F[2,x]+F[66,x]
F[66,x] = F[53,x]
F[67,x] = F[45,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[52,x]+F[71,x]
F[71,x] = F[69,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_{29}\! \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_{23}\! \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_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{63}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{44}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \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_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{28}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \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_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \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_{59}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{49}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{2}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{53}\! \left(x \right)
F_{67}\! \left(x \right) = F_{45}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{52}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{69}\! \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_29(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_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_12(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_19(x) + F_22(x))
Eq(F_22(x), F_12(x)*F_14(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_14(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_12(x) + F_28(x))
Eq(F_28(x), F_26(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_31(x) + F_63(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_44(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_12(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_37(x))
Eq(F_40(x), F_28(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_28(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_18(x) + F_31(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_2(x) + F_45(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_59(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_12(x) + F_56(x))
Eq(F_56(x), F_35(x))
Eq(F_57(x), F_52(x) + F_58(x))
Eq(F_58(x), F_49(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_52(x) + F_60(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), F_2(x) + F_66(x))
Eq(F_66(x), F_53(x))
Eq(F_67(x), F_45(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_52(x) + F_71(x))
Eq(F_71(x), F_69(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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