0132_0231_2031_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 157, 428, 1149, 3058, 8097, 21370, 56279, 147990, 388727, 1020252, 2676139, 7016372, 18389377, 48184544, 126229809, 330635974, 865940277, 2267709166, 5938235819, 15549095466, 40713244907, 106599027888, 279100615999, 730736374568, 1913175616597, 5008924692980, 13113866897829, 34333212871450, 89886845458377, 235329470987362, 616105862471039, 1612996706360382, 4222901436479327, 11055741962816004, 28944393171445459, 75777574990473884, 198388606677883177, 519388794798989576, 1359778877230713369, 3559950035916406126, 9320075628565016157, 24400285645871664598, 63880798901236022099, 167242146242208490578, 437845710194133627347, 1146295125077680746840, 3001039946513885323879, 7856825277413928646160, 20569437011627807457277, 53851488009269307410972, 140985031519779742146189, 369103615557269173768642, 966325833166426288641777, 2529873919970806711120714, 6623295998803587882648359, 17340014220555145012680294, 45396746951092223307104327, 118850227209182277212056236, 311153935829376112935911419, 814611582584789070809372036, 2132680816536677117919592657, 5583430876248614319804181808, 14617611830655909915202504449, 38269404652612603573222434838, 100190602200968877099303006597, 262302402097867980314362997950, 686716604387782969023138813179, 1797847411655776737113759093370, 4706825631760138863035549770427, 12322629485985823093427712824832, 32261062830919696900117233917839, 84460559016218000572663279356152, 221120614236623770749351185005477, 578901283731432243538347437369924, 1515583237033230823591605450523509, 3967848427519375954688297561038954, 10387962045827128495376944526269977, 27196037710566472441249850605124146, 71200151087081214647987236463808719, 186404415553095023141941117135714446, 488013095577039558056294631642159407, 1277634871187695057583859811188413268, 3344891518005388427809118868718379299, 8757039682867155852071164928557322348, 22926227530673450380859712184134783097, 60021642909307937795418644158209417560, 157138701197559848015217565359218250729, 411394460683990576269876742056894896830, 1077044680855649820833698041086365564077, 2819739581885434766309788141752000043942, 7382174064805606238252807905269231064739, 19326782612541287468762918616254886144162, 50598173772838063208664514027893813355427, 132467738706012516238487755636223325897384, 346805042345278713669313017218369708287159
Generating function in Maple syntax:
-(x^6-2*x^5+3*x^4-11*x^3+13*x^2-6*x+1)/(2*x-1)/(x-1)^2/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{x^{6}-2 x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-x**6 + 2*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-1)^2*(x^2-3*x+1)*F(x)+x^6-2*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 -1\right)^{2} \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{6}-2 x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-4*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/20*(4*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)-n-1/4*2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-n -\frac{2^{n}}{4}+1 & \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) = 157
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1-n, 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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 157
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+1-n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19227/
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[40,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[33,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[31,x] = F[32,x]
F[32,x] = F[12,x]*F[30,x]
F[33,x] = F[23,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[37,x] = F[23,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[37,x]
F[40,x] = F[18,x]+F[41,x]+F[71,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[55,x]
F[43,x] = F[44,x]+F[6,x]
F[44,x] = F[18,x]+F[45,x]+F[53,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[10,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[47,x]
F[50,x] = F[17,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[50,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[13,x]
F[55,x] = F[40,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[60,x]+F[69,x]
F[60,x] = F[18,x]+F[61,x]+F[67,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[10,x]+F[64,x]
F[64,x] = F[18,x]+F[21,x]+F[45,x]
F[65,x] = F[60,x]+F[66,x]
F[66,x] = F[57,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[23,x]+F[60,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[59,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[68,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_{40}\! \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_{33}\! \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_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \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_{37}\! \left(x \right) = F_{23}\! \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_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{71}\! \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_{55}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{6}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right)
F_{55}\! \left(x \right) = F_{40}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \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_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{69}\! \left(x \right)
F_{60}\! \left(x \right) = F_{18}\! \left(x \right)+F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)+F_{45}\! \left(x \right)
F_{65}\! \left(x \right) = F_{60}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{57}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{23}\! \left(x \right)+F_{60}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{68}\! \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_40(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_33(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) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_30(x))
Eq(F_33(x), F_23(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))
Eq(F_37(x), F_23(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_18(x) + F_41(x) + F_71(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_55(x))
Eq(F_43(x), F_44(x) + F_6(x))
Eq(F_44(x), F_18(x) + F_45(x) + F_53(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_10(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_47(x))
Eq(F_50(x), F_17(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_50(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_13(x))
Eq(F_55(x), F_40(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_60(x) + F_69(x))
Eq(F_60(x), F_18(x) + F_61(x) + F_67(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_10(x) + F_64(x))
Eq(F_64(x), F_18(x) + F_21(x) + F_45(x))
Eq(F_65(x), F_60(x) + F_66(x))
Eq(F_66(x), F_57(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_23(x) + F_60(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_59(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_68(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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