0231_0321_1203_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 139, 344, 833, 1985, 4675, 10923, 25370, 58644, 135029, 309903, 709319, 1619750, 3691368, 8398048, 19077456, 43281052, 98080402, 222042151, 502239128, 1135150927, 2563920639, 5787596658, 13057650158, 29446350559, 66377503024, 149573203220, 336936552409, 758786791980, 1708372907033, 3845468001358, 8654266013167, 19473114911407, 43810003416065, 98548806893201, 221654404616672, 498487417361650, 1120960041749051, 2520512314847581, 5667015691291509, 12740620530099380, 28641818185478286, 64385388707435574, 144728270065592254, 325312700643826130, 731193462627177536, 1643421715795326069, 3693624913499477718, 8301279520018052245, 18656365117442152613, 41927590600806672624, 94224678317845031284, 211749405156669753837, 475853544105608208766, 1069347107200501824866, 2403028937650863474143, 5400012606998463303266, 12134629381650973733479, 27268087107056918251332, 61274480337581088843293, 137689797098197606025181, 309400744822178350283711, 695246321195491153031991, 1562262619695998920887782, 3510488874927382385654288, 7888230166679416021425153, 17725158825238502472138283, 39829003415525612509090275, 89496824436203459326979266, 201101271355880324407081588, 451877919518811087064394188, 1015375397530044222073574636, 2281557666368509572533829640, 5126673257039233011535575150, 11519649675498892836448876035, 25884656726832432037734438932, 58162773442452369752052061339, 130691521076893970408729450635, 293663093150719261938456053678, 659858802498502891346948556538, 1482696914196076607850341835771, 3331605011990427912362612413580, 7486079084179411675735490584624, 16821128163155138924996996703397, 37796854192503190151394361490392, 84929005051989964628105208924533, 190834231309343694634717742285882, 428801603830205592050366251305419, 963510414621827770715784874571083, 2164992163172643152064097935359357, 4864701059953159909227373848379549, 10930899714089638052670579466337580, 24561540016224798568239472263275630, 55189342069116086691392224979669703, 124009451205427356721293492426598137, 278646957993866047211620036910231569
Generating function in Maple syntax:
(2*x^8-7*x^7+9*x^6+x^5-3*x^4+6*x^3-9*x^2+5*x-1)/(2*x-1)/(x^2+1)/(x^3-x^2-2*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{8}-7 x^{7}+9 x^{6}+x^{5}-3 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(2 x -1\right) \left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**8 - 7*x**7 + 9*x**6 + x**5 - 3*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 + 1)*(x**3 - x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+1)*(x^3-x^2-2*x+1)*(x-1)^2*F(x)-2*x^8+7*x^7-9*x^6-x^5+3*x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-2 x^{8}+7 x^{7}-9 x^{6}-x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/38220*((-865*I*3^(1/2)-125)*(-28+84*I*3^(1/2))^(2/3)-490*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)+7910*(-28+84*I*3^(1/2))^(1/3)-980)*(1/168*(I*3^(1/2)+5)*(-28+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/38220*((370*I*3^(1/2)+1360)*(-28+84*I*3^(1/2))^(2/3)-3710*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-4690*(-28+84*I*3^(1/2))^(1/3)-980)*(1/84*(I*3^(1/2)-2)*(-28+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/38220*((495*I*3^(1/2)-1235)*(-28+84*I*3^(1/2))^(2/3)+4200*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-3220*(-28+84*I*3^(1/2))^(1/3)-980)*(1/6*(-28+84*I*3^(1/2))^(1/3)+1/3-1/168*(-28+84*I*3^(1/2))^(2/3)-1/56*I*3^(1/2)*(-28+84*I*3^(1/2))^(2/3))^(-n)-3/2*n-8/5*2^n-21/65*cos(1/2*n*Pi)+41/130*sin(1/2*n*Pi)+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-865 \,\mathrm{I} \sqrt{3}-125\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-490 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+7910 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-980\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{38220}\\+\\\frac{\left(\left(370 \,\mathrm{I} \sqrt{3}+1360\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-3710 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-4690 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-980\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{38220}\\+\\\frac{\left(\left(495 \,\mathrm{I} \sqrt{3}-1235\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+4200 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-3220 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-980\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}\right)^{-n}}{38220}\\-\frac{3 n}{2}-\frac{8 \,2^{n}}{5}-\frac{21 \cos \left(\frac{n \pi}{2}\right)}{65}+\frac{41 \sin \left(\frac{n \pi}{2}\right)}{130}+2 & \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) = 54
a(6) = 139
a(7) = 344
a(8) = 833
a(n+6) = 2*a(n)-3*a(n+1)-a(n+2)+a(n+3)-4*a(n+4)+4*a(n+5)-3*n-2, 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) = 54
a \! \left(6\right) = 139
a \! \left(7\right) = 344
a \! \left(8\right) = 833
a \! \left(n +6\right) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-3 n -2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20811/
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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[14,x]+F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[58,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[11,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[11,x]+F[30,x]
F[33,x] = F[34,x]+F[42,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[2,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[2,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[19,x]+F[40,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[54,x]
F[48,x] = F[11,x]+F[49,x]
F[49,x] = F[19,x]+F[50,x]+F[52,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[48,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[11,x]
F[54,x] = F[39,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[45,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[37,x]+F[60,x]
F[60,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_{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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \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_{14}\! \left(x \right)+F_{7}\! \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_{58}\! \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_{33}\! \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_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \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_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \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_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)
F_{54}\! \left(x \right) = F_{39}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{45}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{37}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \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_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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_14(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_58(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_11(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_11(x) + F_30(x))
Eq(F_33(x), F_34(x) + F_42(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_2(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_2(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_19(x) + F_40(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_54(x))
Eq(F_48(x), F_11(x) + F_49(x))
Eq(F_49(x), F_19(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_48(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_11(x))
Eq(F_54(x), F_39(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_45(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_37(x) + F_60(x))
Eq(F_60(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:
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