0123_0231_2103_2130
Counting sequence:
1, 1, 2, 6, 20, 61, 166, 423, 1038, 2490, 5892, 13831, 32321, 75343, 175399, 408047, 948938, 2206411, 5129739, 11925714, 27724505, 64452308, 149834165, 348322979, 809753571, 1882449640, 4376165544, 10173351780, 23650174822, 54979987456, 127812965589, 297129097844, 690741351062, 1605778824427, 3732982870432, 8678132315029, 20174210030272, 46899348332912, 109027757255046, 253458785131517, 589220189219411, 1369770754652397, 3184330670652153, 7402670691873461, 17209121488970980, 40006353753820801, 93003488975396661, 216206880907522248, 502620018525597243, 1168449782737147050, 2716316192067772207, 6314669029254623201, 14679824486365475821, 34126451592654956998, 79334374834488546434, 184430045804521205162, 428747839337241483808, 996717801237170591886, 2317087770841550017719, 5386575549387550357874, 12522268907716721634920, 29110743395216614211753, 67674267919603949728490, 157323585876095342402259, 365732965184294741967106, 850225991720297490831030, 1976535630668398330967064, 4594887873748894753212355, 10681818351630185088540277, 24832214938061164090169691, 57727895984672016846647579, 134201076429523907448150539, 311979652257288852741333262, 725264699897490760174353607, 1686035871607418482488552367, 3919557867284562779858290894, 9111866558536342134772129517, 21182519812647319327088367312, 49243384188153836491579142201, 114476979497703212955332830127, 266126689929449285209928582151, 618669494981095266210699237492, 1438232084582090441167573387628, 3343483953713530076291250279616, 7772657186957504612749303310890, 18069235738027544126832982771116, 42005876793881153231291591981481, 97651816092545876052958113723648, 227012930427902865823124139989898, 527741035892497998594747784514935, 1226849062914234140190953187299972, 2852078047385609289206488132871521, 6630277052220857585832305808541396, 15413524124805588319274894347193228, 35832096317360659075366559557380794, 83299517754691658173382195786309977, 193647884754159244688688362591374503, 450176715070455076794667255758897561, 1046533893457738399180007237880267149, 2432896134986464288639375564714394609, 5655797333114371144352779474141561160
Generating function in Maple syntax:
-(x^8-4*x^7+2*x^6+2*x^5-7*x^4+9*x^3-10*x^2+5*x-1)/(x^3-2*x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{8}-4 x^{7}+2 x^{6}+2 x^{5}-7 x^{4}+9 x^{3}-10 x^{2}+5 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**8 + 4*x**7 - 2*x**6 - 2*x**5 + 7*x**4 - 9*x**3 + 10*x**2 - 5*x + 1)/((x - 1)**3*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)+x^8-4*x^7+2*x^6+2*x^5-7*x^4+9*x^3-10*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-4 x^{7}+2 x^{6}+2 x^{5}-7 x^{4}+9 x^{3}-10 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/3450*(69*((-37/69*23^(1/2)+I)*3^(1/2)-37/23*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+3450-690*((I+2/69*23^(1/2))*3^(1/2)-2/23*I*23^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/3450*(690*((-2/69*23^(1/2)+I)*3^(1/2)-2/23*I*23^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+3450-69*((37/69*23^(1/2)+I)*3^(1/2)-37/23*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/3450*((40*2^(2/3)*23^(1/2)*3^(1/2)-1380*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+3450+(74*23^(1/2)*3^(1/2)*2^(1/3)-138*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-3/2*n^2+7/2*n-7)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(69 \left(\left(-\frac{37 \sqrt{23}}{69}+\mathrm{I}\right) \sqrt{3}-\frac{37 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+3450-690 \left(\left(\mathrm{I}+\frac{2 \sqrt{23}}{69}\right) \sqrt{3}-\frac{2 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{3450}\\+\\\frac{\left(690 \left(\left(-\frac{2 \sqrt{23}}{69}+\mathrm{I}\right) \sqrt{3}-\frac{2 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+3450-69 \left(\left(\frac{37 \sqrt{23}}{69}+\mathrm{I}\right) \sqrt{3}-\frac{37 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{3450}\\+\\\frac{\left(\left(40 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-1380 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+3450+\left(74 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}-138 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{3450}\\-\frac{3 n^{2}}{2}+\frac{7 n}{2}-7 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 166
a(7) = 423
a(8) = 1038
a(n+3) = 3/2*n^2+a(n)-2*a(n+1)+3*a(n+2)-1/2*n+5, 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 166
a \! \left(7\right) = 423
a \! \left(8\right) = 1038
a \! \left(n +3\right) = \frac{3 n^{2}}{2}+a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)-\frac{n}{2}+5, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15052/
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[17,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[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[33,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[30,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[27,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[13,x]+F[32,x]
F[32,x] = F[23,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[44,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[19,x]+F[37,x]+F[43,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[11,x]+F[40,x]
F[40,x] = F[19,x]+F[24,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[40,x]+F[7,x]
F[43,x] = F[35,x]*F[4,x]
F[44,x] = F[18,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[14,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_{17}\! \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_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{33}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{43}\! \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_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \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_17(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_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_33(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_30(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x) + F_27(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_13(x) + F_32(x))
Eq(F_32(x), F_23(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_44(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_43(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_11(x) + F_40(x))
Eq(F_40(x), F_19(x) + F_24(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_40(x) + F_7(x))
Eq(F_43(x), F_35(x)*F_4(x))
Eq(F_44(x), F_18(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_14(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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