0132_0213_1230_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 264, 554, 1127, 2244, 4399, 8524, 16371, 31223, 59217, 111800, 210284, 394285, 737343, 1375811, 2562248, 4764041, 8845474, 16403697, 30388436, 56244676, 104019008, 192241289, 355076237, 655496858, 1209545858, 2231010632, 4113676377, 7582747446, 13973571929, 25744647666, 47421756285, 87335416889, 160818050939, 296086895062, 545068263774, 1003312781439, 1846635412649, 3398523501721, 6254146211858, 11508486685947, 21176012475100, 38962683007999, 71686075879510, 131887702707958, 242638286651066, 446376821639263, 821169392454391, 1510615838601324, 2778859972007452, 5111774460232006, 9403077447321851, 17296668313210968, 31816303830895299, 58523789635207992, 107649285433224343, 198009642597017323, 364215505017049157, 669927484096579732, 1232238470111534336, 2266520348675339969, 4168911030734518611, 7668033466822633935, 14104053191384797804, 25941949651395496349, 47715576617208779062, 87764071730048469869, 161425630576317992584, 296911803771299885144, 546112063503056238196, 1004466580123788649813, 1847518087096649197297, 3398141452695113042990, 6250198481585674271574, 11495975105019178850660, 21144504484611831885133, 38890984600170293065706, 71531960164066777579373, 131568251752067984366046, 241992494993789610624449, 445094807890628010018637, 818658954094673798070831, 1505751757417983249223986, 2769514419300365080884818, 5093939531148993982260659, 9369229008100394190021437, 17232720659118776984899493, 31695950199170240766565718, 58297998567760511282602927, 107226829128222703984568112, 197221036298697730325352571, 362746282100398394834638954, 667194824036580552678290346, 1227163237050655850614127470, 2257106114311875694436632619
Generating function in Maple syntax:
-(x^8+2*x^7+6*x^6+2*x^5-3*x^4-3*x^3-2*x^2+3*x-1)/(x^2+x-1)/(x^3+x^2+x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{8}+2 x^{7}+6 x^{6}+2 x^{5}-3 x^{4}-3 x^{3}-2 x^{2}+3 x -1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**8 - 2*x**7 - 6*x**6 - 2*x**5 + 3*x**4 + 3*x**3 + 2*x**2 - 3*x + 1)/((x - 1)**2*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^3+x^2+x-1)*(x-1)^2*F(x)+x^8+2*x^7+6*x^6+2*x^5-3*x^4-3*x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}+2 x^{7}+6 x^{6}+2 x^{5}-3 x^{4}-3 x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/48*(((45*I-11*11^(1/2))*3^(1/2)-33*I*11^(1/2)+45)*(17+3*11^(1/2)*3^(1/2))^(1/3)+120+((81*I+13*11^(1/2))*3^(1/2)-39*I*11^(1/2)-81)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(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)+1/48*(((13*11^(1/2)-81*I)*3^(1/2)+39*I*11^(1/2)-81)*(17+3*11^(1/2)*3^(1/2))^(2/3)+120+((-11*11^(1/2)-45*I)*3^(1/2)+33*I*11^(1/2)+45)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(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)+1/48*((-26*11^(1/2)*3^(1/2)+162)*(17+3*11^(1/2)*3^(1/2))^(2/3)+22*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-90*(17+3*11^(1/2)*3^(1/2))^(1/3)+120)*(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)+1/48*(96*5^(1/2)-240)*(-1/2*5^(1/2)-1/2)^(-n)+1/48*(-96*5^(1/2)-240)*(1/2*5^(1/2)-1/2)^(-n)-5/2*n+11/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(45 i-11 \sqrt{11}\right) \sqrt{3}-33 i \sqrt{11}+45\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+120+\left(\left(81 i+13 \sqrt{11}\right) \sqrt{3}-39 i \sqrt{11}-81\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 i+3 \sqrt{11}\right) \sqrt{3}-9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{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}}{48}\\+\\\frac{\left(\left(\left(13 \sqrt{11}-81 i\right) \sqrt{3}+39 i \sqrt{11}-81\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+120+\left(\left(-11 \sqrt{11}-45 i\right) \sqrt{3}+33 i \sqrt{11}+45\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 i+3 \sqrt{11}\right) \sqrt{3}+9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{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}}{48}\\+\\\frac{\left(\left(-26 \sqrt{11}\, \sqrt{3}+162\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+22 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-90 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+120\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}}{48}\\+\frac{\left(96 \sqrt{5}-240\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{48}+\frac{\left(-96 \sqrt{5}-240\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{48}-\frac{5 n}{2}\\+\frac{11}{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) = 51
a(6) = 120
a(7) = 264
a(8) = 554
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)-5*n+16, 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) = 51
a \! \left(6\right) = 120
a \! \left(7\right) = 264
a \! \left(8\right) = 554
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)-5 n +16, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18461/
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[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[47,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[36,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[29,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[26,x]*F[4,x]
F[29,x] = F[14,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[16,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[26,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[2,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[33,x]+F[7,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[55,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[11,x]+F[54,x]
F[54,x] = F[34,x]
F[55,x] = F[43,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,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_{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_{47}\! \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_{36}\! \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_{29}\! \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_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{2}\! \left(x \right)
F_{40}\! \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_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{34}\! \left(x \right)
F_{55}\! \left(x \right) = F_{43}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \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_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_47(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_36(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) + F_29(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_26(x)*F_4(x))
Eq(F_29(x), F_14(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_16(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_26(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_2(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_33(x) + F_7(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_55(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_11(x) + F_54(x))
Eq(F_54(x), F_34(x))
Eq(F_55(x), F_43(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(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:
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