0132_0231_1302_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 136, 337, 828, 2036, 5016, 12369, 30505, 75224, 185479, 457312, 1127526, 2779982, 6854235, 16899609, 41667208, 102733493, 253296760, 624521200, 1539801480, 3796490201, 9360517005, 23079021456, 56902971403, 140298329348, 345915524746, 852879366454, 2102834830131, 5184689062493, 12783220198680, 31517939972649, 77709726084676, 191598865071500, 472400649780568, 1164737452018801, 2871743154386865, 7080487306795160, 17457445811302719, 43042576174395672, 106124537561454574, 261657606807748094, 645135467946906523, 1590627450433249345, 3921805282421709048, 9669490281361309037, 23840817064636438944, 58781232699004337736, 144929316316919083336, 357333552966563961497, 881031328378673442517, 2172245497632631212592, 5355826007536849522275, 13205170527120262353692, 32558288563694695441170, 80274779641788018563046, 197923187330039384107411, 487993716802301529937925, 1203183269484373254903784, 2966534055916103522247745, 7314201026649452404285836, 18033683635470411770615972, 44463331576382695627777944, 109627511208127080870889425, 270294437857006080656515897, 666430190116537005154950296, 1643131104805498859949889975, 4051256782210328655994356112, 9988662175163431339692825462, 24627758104018578658649191214, 60721491886888143904115705883, 149713163552949976342295343849, 369128469089414921824550419624, 910112534253574237299072076901, 2243947282226072269100422241032, 5532611864904662655379979838496, 13641048651249011987915024379528, 33632977127150951482061465048729, 82924500847072733954810677812765, 204456263706283214372680459146000, 504101481971222042162852842666363, 1242898111894690298652441108373748, 3064453828841500626913541597601178, 7555629201806217140880519376535062, 18628941998701436324911692843255603, 45930983472299161475984799283150637, 113246111501108115702022291045929720, 279216354639040964402900246297732505, 688427811467520763529443927810733332, 1697367807178202579209341352892059964, 4184981235292329392398846511407770712, 10318369339680866964137011064602983025, 25440674603796428632181835760049258433, 62725795422658367885238827251405430744, 154654916690693929272457163399184379887, 381312713460873163560180024619858141128, 940153656657998111534150763237726957310
Generating function in Maple syntax:
-(x^5-2*x^4+3*x^3-6*x^2+4*x-1)/(x-1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
-\frac{x^{5}-2 x^{4}+3 x^{3}-6 x^{2}+4 x -1}{\left(x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(-x**5 + 2*x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/((x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(3*x^3-5*x^2+4*x-1)*F(x)+x^5-2*x^4+3*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{5}-2 x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/810216*(248*((123/124*31^(1/2)+I)*3^(1/2)+369/124*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+600160+9889*((57/899*31^(1/2)+I)*3^(1/2)-171/899*I*31^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3))*(47/8712*2^(1/3)*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/810216*(-9889*2^(2/3)*((-57/899*31^(1/2)+I)*3^(1/2)-171/899*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(1/3)+600160-248*((-123/124*31^(1/2)+I)*3^(1/2)+369/124*I*31^(1/2)-1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)-1+1/810216*((-1254*2^(2/3)*31^(1/2)*3^(1/2)+19778*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+600160+(-492*31^(1/2)*3^(1/2)*2^(1/3)-496*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(248 \left(\left(\frac{123 \sqrt{31}}{124}+\mathrm{I}\right) \sqrt{3}+\frac{369 \,\mathrm{I} \sqrt{31}}{124}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+600160+9889 \left(\left(\frac{57 \sqrt{31}}{899}+\mathrm{I}\right) \sqrt{3}-\frac{171 \,\mathrm{I} \sqrt{31}}{899}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{810216}\\+\\\frac{\left(-9889 \,2^{\frac{2}{3}} \left(\left(-\frac{57 \sqrt{31}}{899}+\mathrm{I}\right) \sqrt{3}-\frac{171 \,\mathrm{I} \sqrt{31}}{899}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+600160-248 \left(\left(-\frac{123 \sqrt{31}}{124}+\mathrm{I}\right) \sqrt{3}+\frac{369 \,\mathrm{I} \sqrt{31}}{124}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{810216}\\-1+\\\frac{\left(\left(-1254 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}+19778 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+600160+\left(-492 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-496 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{810216} & \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) = 53
a(n+3) = 3*a(n)-5*a(n+1)+4*a(n+2)+1, n >= 6
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) = 53
a \! \left(n +3\right) = 3 a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19087/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[20,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[42,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[15,x]+F[22,x]+F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[13,x]
F[29,x] = F[30,x]+F[36,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[34,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[39,x]*F[4,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[2,x]+F[45,x]
F[45,x] = F[22,x]+F[46,x]+F[53,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[11,x]+F[49,x]
F[49,x] = F[15,x]+F[22,x]+F[50,x]
F[50,x] = F[13,x]*F[4,x]
F[51,x] = F[34,x]+F[52,x]
F[52,x] = F[37,x]
F[53,x] = F[4,x]*F[44,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{42}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{37}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_20(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_42(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_15(x) + F_22(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_13(x))
Eq(F_29(x), F_30(x) + F_36(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_34(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_39(x)*F_4(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_2(x) + F_45(x))
Eq(F_45(x), F_22(x) + F_46(x) + F_53(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_11(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_22(x) + F_50(x))
Eq(F_50(x), F_13(x)*F_4(x))
Eq(F_51(x), F_34(x) + F_52(x))
Eq(F_52(x), F_37(x))
Eq(F_53(x), F_4(x)*F_44(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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