0123_0132_1203_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 154, 422, 1139, 3045, 8088, 21388, 56387, 148345, 389700, 1022690, 2681929, 7029643, 18419040, 48249622, 126370557, 330936965, 866578130, 2269050864, 5941040609, 15554928341, 40725321378, 106623936282, 279151822299, 730841342899, 1913390254002, 5009362613826, 13114758642083, 34335025609353, 89890524732232, 235336928485136, 616120959464155, 1613027235092357, 4222963109636716, 11055866441567598, 28944644211824713, 75778080902238783, 198389625547732320, 519390845498889738, 1359783002468041381, 3559958330235111137, 9320092297844204810, 24400319132753397292, 63880866147808670577, 167242281237128104729, 437845981106879711410, 1146295668600663271102, 3001041036682021901587, 7856827463492770542751, 20569441394347721877050, 53851496794136107147566, 140985049125245111884291, 369103650833920885033501, 966325903840609424122216, 2529874061541505437085988, 6623296282353918474322787, 17340014788407951503728257, 45396748088181247191647748, 118850229485904814331033674, 311153940387501232733903609, 814611591709647256217731939, 2132680834802225969458616016, 5583430912808832494976945038, 14617611903829907164177426621, 38269404799059111922618695085, 100190602494043089020266055346, 262302402684349674128535435448, 686716605561359335697346972465, 1797847414004056914702455564621, 4706825636458772911471332490034, 12322629495386905306843811478258, 32261062849728876580992634368827, 84460559053849262998260206393723, 221120614311910028034978901574706, 578901283882048408861926062282346, 1515583237334543440489365880752147, 3967848428122148857921178656168225, 10387962047032945908282993323378424, 27196037712978606829190198221266888, 71200151091906402261873828559542107, 186404415562747088376288734819405037, 488013095596346796931698447387138316, 1277634871226315252585956353411640694, 3344891518082639333477883877767925465, 8757039683021677004731264362370373495, 22926227530982528260418341302811205296, 60021642909926158985761473986929632162, 157138701198796410742689796274790329029, 411394460686463923077071776988588392809, 1077044680860596921578350826899800914890, 2819739581895329716628393573687600093076, 7382174064825397516201831918499758208929, 19326782612580872557932342368334045185009, 50598173772917238046415850267338292583202, 132467738706170874483986105725375877297590, 346805042345595445922989259467842669930963
Generating function in Maple syntax:
-(4*x^4+2*x^3+x^2-3*x+1)/(x^2-3*x+1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{4 x^{4}+2 x^{3}+x^{2}-3 x +1}{\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-4*x**4 - 2*x**3 - x**2 + 3*x - 1)/((x**2 - 3*x + 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3+x^2+x-1)*F(x)+4*x^4+2*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{4}+2 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
1/24*((((-7/132*3^(1/2)-29/132*I)*11^(1/2)+1/4*I*3^(1/2)+1/4)*(17+3*11^(1/2)*3^(1/2))^(1/3)+((5/132*3^(1/2)-49/132*I)*11^(1/2)+3/4*I*3^(1/2)-1/4)*(17+3*11^(1/2)*3^(1/2))^(2/3)+3/2+1/66*I*11^(1/2))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+((-1/2+(3/22*3^(1/2)-1/33*I)*11^(1/2))*(17+3*11^(1/2)*3^(1/2))^(1/3)+((-9/44*3^(1/2)-17/132*I)*11^(1/2)+1/4*I*3^(1/2)+5/4)*(17+3*11^(1/2)*3^(1/2))^(2/3)+3/2-1/66*I*11^(1/2))*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/20*(5^(1/2)-5)*(((-I+1/3*3^(1/2))*11^(1/2)+I*3^(1/2)-1)*(17+3*11^(1/2)*3^(1/2))^(1/3)+16+((-2*I-2/3*3^(1/2))*11^(1/2)+4*I*3^(1/2)+4)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(3/2-1/2*5^(1/2))^(-n)-1/20*(5+5^(1/2))*(((-I+1/3*3^(1/2))*11^(1/2)+I*3^(1/2)-1)*(17+3*11^(1/2)*3^(1/2))^(1/3)+16+((-2*I-2/3*3^(1/2))*11^(1/2)+4*I*3^(1/2)+4)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(3/2+1/2*5^(1/2))^(-n)+(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n))*(((-27/22*I-9/22*3^(1/2))*11^(1/2)+5/2*I*3^(1/2)+5/2)*(17+3*11^(1/2)*3^(1/2))^(2/3)-8+((-9/11*I+3/11*3^(1/2))*11^(1/2)+I*3^(1/2)-1)*(17+3*11^(1/2)*3^(1/2))^(1/3))
Explicit closed form in latex syntax:
\frac{\left(\left(\left(\left(-\frac{7 \sqrt{3}}{132}-\frac{29 \,\mathrm{I}}{132}\right) \sqrt{11}+\frac{\mathrm{I} \sqrt{3}}{4}+\frac{1}{4}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{5 \sqrt{3}}{132}-\frac{49 \,\mathrm{I}}{132}\right) \sqrt{11}+\frac{3 \,\mathrm{I} \sqrt{3}}{4}-\frac{1}{4}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{3}{2}+\frac{\mathrm{I} \sqrt{11}}{66}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+\left(\left(-\frac{1}{2}+\left(\frac{3 \sqrt{3}}{22}-\frac{\mathrm{I}}{33}\right) \sqrt{11}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{9 \sqrt{3}}{44}-\frac{17 \,\mathrm{I}}{132}\right) \sqrt{11}+\frac{\mathrm{I} \sqrt{3}}{4}+\frac{5}{4}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{3}{2}-\frac{\mathrm{I} \sqrt{11}}{66}\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}+\frac{\left(\sqrt{5}-5\right) \left(\left(\left(-\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+16+\left(\left(-2 \,\mathrm{I}-\frac{2 \sqrt{3}}{3}\right) \sqrt{11}+4 \,\mathrm{I} \sqrt{3}+4\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}-\frac{\left(5+\sqrt{5}\right) \left(\left(\left(-\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+16+\left(\left(-2 \,\mathrm{I}-\frac{2 \sqrt{3}}{3}\right) \sqrt{11}+4 \,\mathrm{I} \sqrt{3}+4\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}\right) \left(\left(\left(-\frac{27 \,\mathrm{I}}{22}-\frac{9 \sqrt{3}}{22}\right) \sqrt{11}+\frac{5 \,\mathrm{I} \sqrt{3}}{2}+\frac{5}{2}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-8+\left(\left(-\frac{9 \,\mathrm{I}}{11}+\frac{3 \sqrt{3}}{11}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right)}{24}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = a(n)-2*a(n+1)-a(n+2)-3*a(n+3)+4*a(n+4), 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 +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16402/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[40,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[7,x]
F[28,x] = F[24,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[2,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[20,x]
F[39,x] = F[35,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[33,x]+F[42,x]
F[42,x] = F[20,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \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_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \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_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{20}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_40(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_30(x) + F_35(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) + F_34(x))
Eq(F_33(x), F_2(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_20(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_33(x) + F_42(x))
Eq(F_42(x), F_20(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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