0132_1032_1302_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 58, 169, 477, 1321, 3614, 9796, 26345, 70368, 186841, 493557, 1297967, 3400094, 8875958, 23099327, 59948790, 155194277, 400853893, 1033235153, 2658222226, 6826993092, 17505432341, 44820206532, 114598574101, 292637999557, 746390178319, 1901597759050, 4839712589670, 12305420570683, 31258968232514, 79337263307281, 201198782937101, 509845346098329, 1291024284977206, 3266853656150244, 8261130581525585, 20877472860779192, 52730037322112417, 133104302229886517, 335808191684970927, 846771456682124102, 2134158754028665558, 5376282030654680903, 13537572986303859742, 34073059431783858285, 85723642248380206901, 215583950139280373505, 541957592322908399434, 1361926317543281609956, 3421270002263682548317, 8591563184578538082364, 21568179724883990231421, 54127297762972258666117, 135795556831878902936975, 340585147328088624746194, 853967057090248898446022, 2140601973157788668829603, 5364312108611060333354122, 13439378998950700753331001, 33661616543909868717700541, 84291684076692530047302281, 211023938782595352015952142, 528176459837938370979920900, 1321690088223322110032932857, 3306635499568371209952543440, 8270881828344174718018563241, 20683722170977210958934007541, 51715346834166230230446762031, 129278537361062422614637878318, 323111145669305775161821869302, 807417497314796367577758808335, 2017284643525365701702006084998, 5039193493600419014203866406645, 12585823955152870401391522122085, 31429024575183480560136663879601, 78471251548789708365769659829698, 195894576016086482873484934733188, 488954067532393180686768214207205, 1220250908113415269076156098663156, 3044859001573510540202445103278629, 7596677608628175902361938132566725, 18950475308787456808396091521169935, 47266975062618234745080848652564378, 117879462027876995427181522739543078, 293942517850568832800223306638108747, 732878290074285878166413553404878610, 1827038492479515780797516219133526049, 4554189553764346878008615491543157997, 11350714679659388061887301710543887097, 28286883857085975578567958362019509926, 70485188568780854998582237137860885924, 175615365559607229290712463883145767905, 437501999991815230257290665091123103528, 1089811594957342927039295720262267819761, 2714423419338862856867105224064154093621, 6760198675372384978255968418489440558511, 16834410138276663092437505847100430733910
Generating function in Maple syntax:
(x^2-x+1)*(2*x^5+2*x^4-11*x^3+13*x^2-6*x+1)/(3*x^3-5*x^2+4*x-1)^2
Generating function in latex syntax:
\frac{\left(x^{2}-x +1\right) \left(2 x^{5}+2 x^{4}-11 x^{3}+13 x^{2}-6 x +1\right)}{\left(3 x^{3}-5 x^{2}+4 x -1\right)^{2}}
Generating function in sympy syntax:
(x**2 - x + 1)*(2*x**5 + 2*x**4 - 11*x**3 + 13*x**2 - 6*x + 1)/(3*x**3 - 5*x**2 + 4*x - 1)**2
Implicit equation for the generating function in Maple syntax:
-(3*x^3-5*x^2+4*x-1)^2*F(x)+(x^2-x+1)*(2*x^5+2*x^4-11*x^3+13*x^2-6*x+1) = 0
Implicit equation for the generating function in latex syntax:
-\left(3 x^{3}-5 x^{2}+4 x -1\right)^{2} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(2 x^{5}+2 x^{4}-11 x^{3}+13 x^{2}-6 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/75350088*(11594*(((-15/17*n-429/527)*31^(1/2)+I*n-220/17*I)*3^(1/2)+(45/17*I*n+1287/527*I)*31^(1/2)-n+220/17)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)-37634*(((-39/607*n+3747/37634)*31^(1/2)+I*n+113/1214*I)*3^(1/2)+(-117/607*I*n+11241/37634*I)*31^(1/2)+n+113/1214)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+5341424*n+6511736)*(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)+1/75350088*(-11594*(((15/17*n+429/527)*31^(1/2)+I*n-220/17*I)*3^(1/2)+(45/17*I*n+1287/527*I)*31^(1/2)+n-220/17)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+37634*2^(1/3)*(((39/607*n-3747/37634)*31^(1/2)+I*n+113/1214*I)*3^(1/2)+(-117/607*I*n+11241/37634*I)*31^(1/2)-n-113/1214)*(47+9*31^(1/2)*3^(1/2))^(2/3)+5341424*n+6511736)*(-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)-13/202554*(-55/13*2^(2/3)*(3^(1/2)*(n+143/155)*31^(1/2)+17/15*n-44/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+(3^(1/2)*(n-1249/806)*31^(1/2)-607/39*n-113/78)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)-43076/39*n-52514/39)*(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))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(11594 \left(\left(\left(-\frac{15 n}{17}-\frac{429}{527}\right) \sqrt{31}+\mathrm{I} n -\frac{220 \,\mathrm{I}}{17}\right) \sqrt{3}+\left(\frac{45 \,\mathrm{I} n}{17}+\frac{1287 \,\mathrm{I}}{527}\right) \sqrt{31}-n +\frac{220}{17}\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-37634 \left(\left(\left(-\frac{39 n}{607}+\frac{3747}{37634}\right) \sqrt{31}+\mathrm{I} n +\frac{113 \,\mathrm{I}}{1214}\right) \sqrt{3}+\left(-\frac{117 \,\mathrm{I} n}{607}+\frac{11241 \,\mathrm{I}}{37634}\right) \sqrt{31}+n +\frac{113}{1214}\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+5341424 n +6511736\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}}{75350088}\\+\\\frac{\left(-11594 \left(\left(\left(\frac{15 n}{17}+\frac{429}{527}\right) \sqrt{31}+\mathrm{I} n -\frac{220 \,\mathrm{I}}{17}\right) \sqrt{3}+\left(\frac{45 \,\mathrm{I} n}{17}+\frac{1287 \,\mathrm{I}}{527}\right) \sqrt{31}+n -\frac{220}{17}\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+37634 \,2^{\frac{1}{3}} \left(\left(\left(\frac{39 n}{607}-\frac{3747}{37634}\right) \sqrt{31}+\mathrm{I} n +\frac{113 \,\mathrm{I}}{1214}\right) \sqrt{3}+\left(-\frac{117 \,\mathrm{I} n}{607}+\frac{11241 \,\mathrm{I}}{37634}\right) \sqrt{31}-n -\frac{113}{1214}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+5341424 n +6511736\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}}{75350088}\\-\\\frac{13 \left(-\frac{55 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n +\frac{143}{155}\right) \sqrt{31}+\frac{17 n}{15}-\frac{44}{3}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}+\left(\sqrt{3}\, \left(n -\frac{1249}{806}\right) \sqrt{31}-\frac{607 n}{39}-\frac{113}{78}\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{43076 n}{39}-\frac{52514}{39}\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}}{202554} & \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) = 58
a(6) = 169
a(7) = 477
a(n+6) = -9*a(n)+30*a(n+1)-49*a(n+2)+46*a(n+3)-26*a(n+4)+8*a(n+5), 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) = 58
a \! \left(6\right) = 169
a \! \left(7\right) = 477
a \! \left(n +6\right) = -9 a \! \left(n \right)+30 a \! \left(n +1\right)-49 a \! \left(n +2\right)+46 a \! \left(n +3\right)-26 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19770/
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[28,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[16,x]+F[27,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[22,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[21,x]*F[4,x]
F[27,x] = F[13,x]*F[4,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[43,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[11,x]+F[14,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[2,x]+F[46,x]
F[46,x] = F[15,x]+F[30,x]+F[47,x]
F[47,x] = F[4,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_{28}\! \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_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{43}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \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_{2}\! \left(x \right)+F_{34}\! \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_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \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_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \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_28(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) + F_16(x) + F_27(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_22(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_21(x)*F_4(x))
Eq(F_27(x), F_13(x)*F_4(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_43(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_11(x) + F_14(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_2(x) + F_46(x))
Eq(F_46(x), F_15(x) + F_30(x) + F_47(x))
Eq(F_47(x), F_4(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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