0231_1203_1302_2301_3012
Counting sequence:
1, 1, 2, 6, 19, 56, 153, 396, 991, 2433, 5915, 14321, 34644, 83888, 203501, 494723, 1205263, 2942054, 7193908, 17616344, 43191794, 106006060, 260391212, 640062745, 1574217552, 3873537653, 9534864573, 23477647840, 57823329698, 142442567451, 350952350464, 864798121978, 2131221571181, 5252678988988, 12946854987387, 31913394756408, 78668751127153, 193931415826197, 478087732207125, 1178631383610117, 2905743681938510, 7163806128427140, 17661850499042471, 43544502888232719, 107357978419755993, 264690552688489154, 652597031226865974, 1608991704538256012, 3967018135712088124, 9780834745093509048, 24115082677940025718, 59456929918452129917, 146594247569435516658, 361436462819394672553, 891143151385943833679, 2197168530555481175012, 5417258746551134833284, 13356603773186455503527, 32931590922859572436642, 81195049007565378830278, 200191908620334943823639, 493586873981715446774032, 1216972523387130901079677, 3000530296579446595108660, 7398016885484269001702403, 18240329017515217307157337, 44972813309005827150709359, 110883640358155955362650217, 273391445046652190199468808, 674065944535870170598593864, 1661957326410735007808619457, 4097673622955640569487141675, 10103105103080687391995654243, 24909923096185131026561520910, 61417185377020767419906690456, 151428436614032993853160968720, 373357579414850565653504622982, 920539671830833664129352932660, 2269656062312248946852209882848, 5595998552781515437808265391281, 13797332763941590557531424186900, 34018306176564076977983705842509, 83874555940429991630387911338689, 206798689961798533689598214082296, 509876896256884768311146067839718, 1257137798204133101229240236230883, 3099562771746097607512991850417956, 7642192765391566115005137628836770, 18842370558762549498475178381454913, 46457206645959407786077733751800660, 114543551931457077086277921624157855, 282415285862833908088803484157325632, 696315003112958450639367392314470869, 1716814237694095454086905638444726413, 4232927790324211208846435494880405609, 10436584977213482759727794864645036125, 25732143660411641042656676700319436226, 63444433106744743369960954564307932220, 156426769016947135288611650552797824123, 385681341436071584918347521897066141463, 950924819821328568311686023330111493613
Generating function in Maple syntax:
-(11*x^7-43*x^6+85*x^5-101*x^4+76*x^3-35*x^2+9*x-1)/(2*x-1)/(3*x^3-5*x^2+4*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{11 x^{7}-43 x^{6}+85 x^{5}-101 x^{4}+76 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-11*x**7 + 43*x**6 - 85*x**5 + 101*x**4 - 76*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**4*(2*x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(3*x^3-5*x^2+4*x-1)*(x-1)^4*F(x)+11*x^7-43*x^6+85*x^5-101*x^4+76*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+11 x^{7}-43 x^{6}+85 x^{5}-101 x^{4}+76 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
1/45012*(-341*((-15/31*I+5/31*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+15004-124*2^(1/3)*((3/62*I+1/62*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*(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)+1/45012*(341*((-15/31*I-5/31*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+15004+124*2^(1/3)*((3/62*I-1/62*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-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)+1/45012*((110*31^(1/2)*3^(1/2)*2^(2/3)-682*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+15004+(4*31^(1/2)*3^(1/2)*2^(1/3)+248*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)-1/6*n^3-11/6*n+2^n-1
Explicit closed form in latex syntax:
\frac{\left(-341 \left(\left(-\frac{15 \,\mathrm{I}}{31}+\frac{5 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+15004-124 \,2^{\frac{1}{3}} \left(\left(\frac{3 \,\mathrm{I}}{62}+\frac{\sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) \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}}{45012}+\frac{\left(341 \left(\left(-\frac{15 \,\mathrm{I}}{31}-\frac{5 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+15004+124 \,2^{\frac{1}{3}} \left(\left(\frac{3 \,\mathrm{I}}{62}-\frac{\sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{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}}{45012}+\frac{\left(\left(110 \sqrt{31}\, \sqrt{3}\, 2^{\frac{2}{3}}-682 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+15004+\left(4 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+248 \,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}}{45012}-\frac{n^{3}}{6}-\frac{11 n}{6}+2^{n}-1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 153
a(7) = 396
a(n+4) = -6*a(n)+13*a(n+1)-13*a(n+2)+6*a(n+3)-1/6*(n-1)*(n^2-2*n+12), n >= 8
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) = 153
a \! \left(7\right) = 396
a \! \left(n +4\right) = -6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{\left(n -1\right) \left(n^{2}-2 n +12\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21052/
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[13,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[14,x]+F[6,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[20,x]+F[24,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[21,x]+F[6,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[10,x]+F[25,x]
F[28,x] = F[29,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[21,x]+F[28,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[15,x]+F[33,x]+F[37,x]
F[33,x] = F[13,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[6,x]
F[36,x] = F[32,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[20,x]+F[39,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[15,x]+F[41,x]+F[43,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[10,x]+F[40,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[10,x]+F[40,x]
F[45,x] = F[46,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[21,x]+F[45,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_31(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_14(x) + F_6(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_24(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_21(x) + F_6(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_25(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_21(x) + F_28(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_15(x) + F_33(x) + F_37(x))
Eq(F_33(x), F_13(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_6(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_20(x) + F_39(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_15(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_10(x) + F_40(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_40(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_21(x) + F_45(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, 2, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 2, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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