0132_1203_1230_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 133, 314, 707, 1540, 3276, 6842, 14087, 28675, 57827, 115714, 230035, 454742, 894598, 1752462, 3420155, 6652753, 12902337, 24956090, 48154515, 92714072, 178149968, 341688290, 654248463, 1250777855, 2387771583, 4552242114, 8667983843, 16485690986, 31320246858, 59443137302, 112710314083, 213518482669, 404147172989, 764357231994, 1444524173219, 2727986364908, 5148308104596, 9709718049418, 18301310046743, 34474973469691, 64905835769243, 122132888285762, 229699937997043, 431795505727806, 811322185158734, 1523754566179742, 2860559891186251, 5367955067088905, 10069212520435129, 18880676533374586, 35390054596107251, 66312046175770048, 124210039360977816, 232583715189642674, 435376740364757023, 814740271666257399, 1524213018666690935, 2850683038532985474, 5330071404898127555, 9963267837411311570, 18619110740023405970, 34786273892859527718, 64975865215315416755, 121337374962927185829, 226536675841378819701, 422850415649648118458, 789118252968986344835, 1472336792375348889300, 2746531195054377875804, 5122437208889475460186, 9551813346785694704935, 17807966603747197393907, 33194361131397271741779, 61863978057389209855874, 115275471592985278542419, 214764957529657309888806, 400054556703827993601814, 745085447898603926647214, 1387476720475904319815323, 2583329095018080869670593, 4809165853748282180030897, 8951530387881520152698170, 16659571015582573679084883, 31000646245142012335578152, 57679231034109685750991840, 107302856347202178149734786, 199594004052254998527779375, 371218253297239470687690991, 690331954038538284299944495, 1283614484019348121000813634, 2386493966191955488667772067, 4436456792059828511067991098, 8246381177549263718000759258, 15326493533725995544779488566, 28482325424348153815254848259
Generating function in Maple syntax:
-(x^9+2*x^8+2*x^7+3*x^6+x^5-5*x^4-2*x^3-2*x^2+3*x-1)/(x^3+x^2+x-1)^2/(x-1)^2
Generating function in latex syntax:
-\frac{x^{9}+2 x^{8}+2 x^{7}+3 x^{6}+x^{5}-5 x^{4}-2 x^{3}-2 x^{2}+3 x -1}{\left(x^{3}+x^{2}+x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**9 - 2*x**8 - 2*x**7 - 3*x**6 - x**5 + 5*x**4 + 2*x**3 + 2*x**2 - 3*x + 1)/((x - 1)**2*(x**3 + x**2 + x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)^2*(x-1)^2*F(x)+x^9+2*x^8+2*x^7+3*x^6+x^5-5*x^4-2*x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+2 x^{8}+2 x^{7}+3 x^{6}+x^{5}-5 x^{4}-2 x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/11616*((-1254*(I+1/3*3^(1/2))*(n-544/209)*11^(1/2)+1914*(1+I*3^(1/2))*(n-76/29))*(17+3*11^(1/2)*3^(1/2))^(1/3)+(-1023*(I-1/3*3^(1/2))*(n-862/341)*11^(1/2)+2211*(I*3^(1/2)-1)*(n-170/67))*(17+3*11^(1/2)*3^(1/2))^(2/3)+3168*n-8712)*(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)+1/11616*((1254*(I-1/3*3^(1/2))*(n-544/209)*11^(1/2)-1914*(I*3^(1/2)-1)*(n-76/29))*(17+3*11^(1/2)*3^(1/2))^(1/3)+(1023*(I+1/3*3^(1/2))*(n-862/341)*11^(1/2)-2211*(1+I*3^(1/2))*(n-170/67))*(17+3*11^(1/2)*3^(1/2))^(2/3)+3168*n-8712)*(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)+1/11616*(((836*n-2176)*3^(1/2)*11^(1/2)-3828*n+10032)*(17+3*11^(1/2)*3^(1/2))^(1/3)+((-682*n+1724)*3^(1/2)*11^(1/2)+4422*n-11220)*(17+3*11^(1/2)*3^(1/2))^(2/3)+3168*n-8712)*(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)-1/2*n+21/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(-1254 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{544}{209}\right) \sqrt{11}+1914 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{76}{29}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-1023 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{862}{341}\right) \sqrt{11}+2211 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{170}{67}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+3168 n -8712\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{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}}{11616}\\+\\\frac{\left(\left(1254 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{544}{209}\right) \sqrt{11}-1914 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{76}{29}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(1023 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{862}{341}\right) \sqrt{11}-2211 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{170}{67}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+3168 n -8712\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{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}}{11616}\\+\\\frac{\left(\left(\left(836 n -2176\right) \sqrt{3}\, \sqrt{11}-3828 n +10032\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-682 n +1724\right) \sqrt{3}\, \sqrt{11}+4422 n -11220\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+3168 n -8712\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}}{11616}\\-\frac{n}{2}+\frac{21}{4} & \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) = 53
a(6) = 133
a(7) = 314
a(8) = 707
a(9) = 1540
a(n+2) = -1/3*a(n)-2/3*a(n+1)+1/3*a(n+4)+2/3*a(n+5)-1/3*a(n+6)-2/3*n+7, n >= 10
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) = 53
a \! \left(6\right) = 133
a \! \left(7\right) = 314
a \! \left(8\right) = 707
a \! \left(9\right) = 1540
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+\frac{2 a \! \left(n +5\right)}{3}-\frac{a \! \left(n +6\right)}{3}-\frac{2 n}{3}+7, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19829/
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[24,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[22,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[22,x] = F[23,x]*F[4,x]
F[23,x] = F[7,x]
F[24,x] = F[2,x]+F[25,x]
F[25,x] = F[15,x]+F[26,x]+F[47,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[36,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[15,x]+F[16,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[23,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[18,x]
F[36,x] = F[37,x]+F[41,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[2,x]
F[41,x] = F[42,x]
F[42,x] = F[4,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[40,x]+F[49,x]
F[49,x] = F[44,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_{24}\! \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_{22}\! \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_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{47}\! \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_{36}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \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_{18}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{41}\! \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_{2}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \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_{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_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{44}\! \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_24(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_22(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))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_7(x))
Eq(F_24(x), F_2(x) + F_25(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_47(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_36(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_15(x) + F_16(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_23(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_18(x))
Eq(F_36(x), F_37(x) + F_41(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_2(x))
Eq(F_41(x), F_42(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_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_40(x) + F_49(x))
Eq(F_49(x), F_44(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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