0132_0213_0321_1320_3120
Counting sequence:
1, 1, 2, 6, 19, 57, 161, 436, 1151, 2996, 7744, 19960, 51422, 132579, 342296, 885183, 2292877, 5948511, 15454487, 40202110, 104693261, 272894822, 711894262, 1858331746, 4853664044, 12682737717, 33152606366, 86687001321, 226723848571, 593098668801, 1551766851869, 4060524165640, 10626315984587, 27811176031304, 72792179724316, 190534224629308, 498746069654762, 1305570840349431, 3417691573487252, 8946936944429955, 23421951028698841, 61316510959981587, 160522634048921747, 420241220704227586, 1100180137342834169, 2880256310370792602, 7540500832839322546, 19741065868240220806, 51682327335974408216, 135305159675683095081, 354231603578702969594, 927386484466937820045, 2427921375897646146727, 6356364413902095220629, 16641144844210875293561, 43567014949635095372908, 114059787414703726564247, 298612117610895088426076, 781776097043620292183992, 2046715218756844785582400, 5358367613671875040510646, 14028383659091108249914827, 36726775293150917461306832, 96151925791230203486438151, 251728968645816059399447269, 659034912123849202907934087, 1725375629375150996502719951, 4517091694688756662529565766, 11825898882842052706089879461, 30960603791692524812038323086, 81055910131052280295202485294, 212207121805310856909085715050, 555565445544999419513411409092, 1454489195054777754614509182429, 3807902099479218739938131824406, 9969217021922056635698506357233, 26099748801004124266719804770707, 68330029045802295880716097785321, 178890337656381989842199577815957, 468340982544412660647946233266320, 1226132607181215034242809155478051, 3210056833332392662636906976735792, 8404037881331167667328934615023892, 22002056787389288311768285255245316, 57602132433688590303005383337174594, 150804340418171342847692159954494047, 394810888627397307101730428573322956, 1033628325072328612902359273160677451, 2706074085796533193938150652301461681, 7084593930711817480463865139733909691, 18547707703089326644329321545290070219, 48558529171979605993820516785734707530, 127127879799541635914814390856748471057, 332825110199720105896165634805454757954, 871347450745149320045125781464054341226, 2281217241925551190742812865120686214158, 5972304274808675045111944364416161945392, 15635695582049863770293141808064518409041, 40934782470429792396853441177451637857522, 107168651827397458641810535875240498322965, 280571173008038859890407742581373291450431
Generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x^5-10*x^3+13*x^2-6*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{5}-10 x^{3}+13 x^{2}-6 x +1\right)}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**3 - 2*x**2 + 3*x - 1)*(x**5 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**3*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2*(x-1)^3*F(x)+(x^3-2*x^2+3*x-1)*(x^5-10*x^3+13*x^2-6*x+1) = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{5}-10 x^{3}+13 x^{2}-6 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/8*2^n*n-1/2*n^2-1/4*2^n+1/2*n-1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{2^{n} n}{8}-\frac{n^{2}}{2}-\frac{2^{n}}{4}+\frac{n}{2}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5} & \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) = 57
a(6) = 161
a(7) = 436
a(8) = 1151
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+1/2*(n+1)*(n-4), n >= 9
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) = 57
a \! \left(6\right) = 161
a \! \left(7\right) = 436
a \! \left(8\right) = 1151
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{\left(n +1\right) \left(n -4\right)}{2}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21607/
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[16,x]*F[4,x]
F[4,x] = F[40,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[20,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[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[16,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[17,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[25,x]+F[26,x]+F[28,x]
F[25,x] = 0
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[10,x]+F[24,x]
F[28,x] = F[16,x]*F[29,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[21,x]+F[31,x]
F[31,x] = F[25,x]+F[32,x]+F[34,x]
F[32,x] = F[16,x]*F[33,x]
F[33,x] = F[14,x]+F[31,x]
F[34,x] = F[16,x]*F[30,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]
F[37,x] = F[16,x]*F[21,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[35,x]
F[40,x] = F[41,x]
F[41,x] = F[16,x]*F[42,x]
F[42,x] = F[43,x]+F[49,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[45,x]+F[5,x]
F[45,x] = F[14,x]*F[46,x]
F[46,x] = F[1,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[16,x]*F[23,x]*F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[13,x]*F[16,x]*F[51,x]
F[51,x] = F[52,x]+F[69,x]
F[52,x] = F[53,x]+F[68,x]
F[53,x] = F[54,x]+F[59,x]
F[54,x] = F[13,x]*F[46,x]*F[55,x]
F[55,x] = F[13,x]+F[56,x]+F[58,x]
F[56,x] = F[57,x]
F[57,x] = F[13,x]*F[16,x]*F[55,x]
F[58,x] = F[16,x]*F[55,x]
F[59,x] = F[60,x]*F[66,x]
F[60,x] = F[13,x]+F[61,x]
F[61,x] = F[21,x]+F[62,x]
F[62,x] = F[25,x]+F[63,x]+F[65,x]
F[63,x] = F[16,x]*F[64,x]
F[64,x] = F[14,x]+F[62,x]
F[65,x] = F[16,x]*F[61,x]
F[66,x] = F[67,x]
F[67,x] = F[14,x]*F[16,x]*F[21,x]*F[23,x]
F[68,x] = F[13,x]*F[21,x]*F[60,x]
F[69,x] = F[60,x]*F[70,x]
F[70,x] = F[32,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{40}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{20}\! \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_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right) F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{5}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{69}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{68}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{16}\! \left(x \right) F_{55}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right) F_{66}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{21}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{16}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{16}\! \left(x \right) F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{68}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{60}\! \left(x \right)
F_{69}\! \left(x \right) = F_{60}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{32}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_40(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_16(x)*F_8(x))
Eq(F_8(x), F_20(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_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_16(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_17(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_28(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_24(x))
Eq(F_28(x), F_16(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_35(x))
Eq(F_30(x), F_21(x) + F_31(x))
Eq(F_31(x), F_25(x) + F_32(x) + F_34(x))
Eq(F_32(x), F_16(x)*F_33(x))
Eq(F_33(x), F_14(x) + F_31(x))
Eq(F_34(x), F_16(x)*F_30(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_16(x)*F_21(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_35(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_16(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_49(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_5(x))
Eq(F_45(x), F_14(x)*F_46(x))
Eq(F_46(x), F_1(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_16(x)*F_23(x)*F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_13(x)*F_16(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_69(x))
Eq(F_52(x), F_53(x) + F_68(x))
Eq(F_53(x), F_54(x) + F_59(x))
Eq(F_54(x), F_13(x)*F_46(x)*F_55(x))
Eq(F_55(x), F_13(x) + F_56(x) + F_58(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_13(x)*F_16(x)*F_55(x))
Eq(F_58(x), F_16(x)*F_55(x))
Eq(F_59(x), F_60(x)*F_66(x))
Eq(F_60(x), F_13(x) + F_61(x))
Eq(F_61(x), F_21(x) + F_62(x))
Eq(F_62(x), F_25(x) + F_63(x) + F_65(x))
Eq(F_63(x), F_16(x)*F_64(x))
Eq(F_64(x), F_14(x) + F_62(x))
Eq(F_65(x), F_16(x)*F_61(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_14(x)*F_16(x)*F_21(x)*F_23(x))
Eq(F_68(x), F_13(x)*F_21(x)*F_60(x))
Eq(F_69(x), F_60(x)*F_70(x))
Eq(F_70(x), F_32(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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