0123_0231_1032_2013_2130
Counting sequence:
1, 1, 2, 6, 19, 51, 127, 306, 725, 1702, 3976, 9265, 21563, 50155, 116626, 271155, 630394, 1465526, 3406975, 7920299, 18412507, 42803934, 99507125, 231326054, 537767888, 1250158725, 2906266499, 6756249983, 15706375726, 36512893763, 84882179874, 197327127878, 458729917707, 1066417677299, 2479120324423, 5763255536442, 13397943637845, 31146440165142, 72406688756248, 168325129576377, 391308451381851, 909681783749123, 2114753578060122, 4916205618064051, 11428791481821114, 26568716787395446, 61764773016608247, 143585676956855051, 333796201624744195, 775982023977130822, 1803939345638759221, 4193650190586760310, 9749053904459893408, 22663800677844918925, 52686944415201730371, 122482285794375246775, 284736769230567198614, 661932680518152832771, 1538806788887699347970, 3577291774857359577094, 8316194427314832868227, 19332806521117478798579, 44943322483580130236495, 104480548835820265980674, 242887808061418016267733, 564645648996193647078470, 1312641879701565174680744, 3051522149173726246153153, 7093928337114242036176571, 16491382592696838790904283, 38337813253035758546512994, 89124602910827840093907123, 207189564819108841979599802, 481657301888706604297498294, 1119717378938729969027202543, 2603027097857885540466210987, 6051303837584903287641726315, 14067574695977668751019959662, 32703143510621085218242637493, 76025584977492821440329719622, 176738042607213962635523843696, 410866101377277330244154729493, 955147803894896886901746220867, 2220449251537349962852453047471, 5161918248199533444998021430334, 12000004045418797296190904417091, 27896624891394674961429123438242, 64851784831545963736903582911046, 150762108757267338584043406273915, 350479381500104763239752176438067, 814765711817325576288073299677591, 1894100481209034540968758952430730, 4403249401492557233569882434375253, 10236312953876928195060202697942070, 23796540539854704659009602177506616, 55320245113302814820478283571011145, 128603967214075963338475849055962459, 298967951955476965033480582203371891, 695016166551581783243968332069202090, 1615716562957867383003419680856825139, 3756085307725915547555802960635443322
Generating function in Maple syntax:
-(2*x^6-3*x^5+3*x^4-3*x^3+6*x^2-4*x+1)/(x^3-2*x^2+3*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{6}-3 x^{5}+3 x^{4}-3 x^{3}+6 x^{2}-4 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**6 + 3*x**5 - 3*x**4 + 3*x**3 - 6*x**2 + 4*x - 1)/((x - 1)**2*(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)^2*F(x)+2*x^6-3*x^5+3*x^4-3*x^3+6*x^2-4*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)^{2} F \! \left(x \right)+2 x^{6}-3 x^{5}+3 x^{4}-3 x^{3}+6 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/13800*(161*((I-3/161*23^(1/2))*3^(1/2)-9/161*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+18400+115*((I+9/23*23^(1/2))*3^(1/2)-27/23*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)+1/13800*(-115*2^(2/3)*((I-9/23*23^(1/2))*3^(1/2)-27/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+18400-161*((I+3/161*23^(1/2))*3^(1/2)-9/161*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(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)+1/13800*((-90*3^(1/2)*2^(2/3)*23^(1/2)+230*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+18400+(6*23^(1/2)*3^(1/2)*2^(1/3)-322*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)-2*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(161 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{161}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{161}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+18400+115 \left(\left(\mathrm{I}+\frac{9 \sqrt{23}}{23}\right) \sqrt{3}-\frac{27 \,\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}}{13800}\\+\\\frac{\left(-115 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{23}}{23}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+18400-161 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{161}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{161}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \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}}{13800}\\+\\\frac{\left(\left(-90 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}+230 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+18400+\left(6 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}-322 \,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}}{13800}\\-2 n +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) = 127
a(n+3) = a(n)-2*a(n+1)+3*a(n+2)+2*n, n >= 7
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) = 127
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17253/
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[19,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[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[15,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[36,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[7,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[25,x]
F[35,x] = F[32,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[21,x]+F[40,x]+F[50,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[4,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[7,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[4,x]
F[49,x] = F[46,x]
F[50,x] = F[38,x]*F[4,x]
F[51,x] = F[20,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[46,x]
F[56,x] = F[53,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_{19}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \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_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)
F_{35}\! \left(x \right) = F_{32}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \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_{7}\! \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_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{46}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \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_19(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_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_15(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_36(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_31(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_7(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_25(x))
Eq(F_35(x), F_32(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_21(x) + F_40(x) + F_50(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_4(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_7(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_4(x))
Eq(F_49(x), F_46(x))
Eq(F_50(x), F_38(x)*F_4(x))
Eq(F_51(x), F_20(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_46(x))
Eq(F_56(x), F_53(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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