0123_0132_1230_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 50, 111, 245, 525, 1100, 2273, 4639, 9375, 18794, 37419, 74069, 145881, 286060, 558781, 1087783, 2111147, 4086066, 7888935, 15196965, 29215077, 56058844, 107382681, 205369375, 392193335, 747952394, 1424617315, 2710254693, 5150425265, 9777491996, 18543459189, 35136459271, 66519975235, 125832828530, 237849845471, 449258731285, 847991004605, 1599565845164, 3015387526225, 5681032184287, 10697111572943, 20131317054186, 37866460407707, 71190800419125, 133779274632329, 251280143191052, 471780434109997, 885404372284071, 1661003293535195, 3114821180097522, 5838964790385815, 10941696632605893, 20496758996313877, 38383340125586620, 71855899222599433, 134477297912098655, 251597860002257511, 470587782914620170, 879942288916212435, 1644945328439963589, 3074228870742570337, 5743966703264630076, 10729561984691695333, 20037782326581084935, 37412607079830014579, 69837693306522119602, 130337145461527337295, 243195546577185518645, 453685290738554465645, 846190051648486977036, 1577965963958072154049, 2942023355603497935007, 5484228564333506252543, 10221344201270755059050, 19046953680965271488779, 35487059516826172946453, 66106374346452030599993, 123125295121647458529196, 229289186579976116821405, 426927238167919875296039, 794803467161842158231819, 1479458478916931582025650, 2753495900666030024160519, 5123964897463632312286565, 9533871631191952306624709, 17736798550605137987699356, 32993260605408432887258745, 61364974788782115195010463, 114120169593803511708671895, 212203210164726187723720202, 394539339374628360212312643, 733463644786214558801097253, 1363382909982674279411211793, 2534014520280995415840005980, 4709267342497524143298173525, 8750876382003417117884715783, 16259404747609278063019462563
Generating function in Maple syntax:
-(x^9+5*x^8+4*x^7-x^6-11*x^5-8*x^4-x^3+x^2+x-1)/(x^3+x^2+x-1)^2
Generating function in latex syntax:
-\frac{x^{9}+5 x^{8}+4 x^{7}-x^{6}-11 x^{5}-8 x^{4}-x^{3}+x^{2}+x -1}{\left(x^{3}+x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(-x**9 - 5*x**8 - 4*x**7 + x**6 + 11*x**5 + 8*x**4 + x**3 - x**2 - x + 1)/(x**3 + x**2 + x - 1)**2
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)^2*F(x)+x^9+5*x^8+4*x^7-x^6-11*x^5-8*x^4-x^3+x^2+x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right)^{2} F \! \left(x \right)+x^{9}+5 x^{8}+4 x^{7}-x^{6}-11 x^{5}-8 x^{4}-x^{3}+x^{2}+x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 1,1,n = 2,2,n = 3,6,1/5808*((-330*(n+39/11)*(I+1/3*3^(1/2))*11^(1/2)+374*(n+191/17)*(1+I*3^(1/2)))*(17+3*11^(1/2)*3^(1/2))^(1/3)+(-561*(I-1/3*3^(1/2))*(n-1494/187)*11^(1/2)+1133*(n-746/103)*(I*3^(1/2)-1))*(17+3*11^(1/2)*3^(1/2))^(2/3)+352*n+1936)*(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)+1/5808*((330*(I-1/3*3^(1/2))*(n+39/11)*11^(1/2)-374*(I*3^(1/2)-1)*(n+191/17))*(17+3*11^(1/2)*3^(1/2))^(1/3)+(561*(n-1494/187)*(I+1/3*3^(1/2))*11^(1/2)-1133*(n-746/103)*(1+I*3^(1/2)))*(17+3*11^(1/2)*3^(1/2))^(2/3)+352*n+1936)*(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)+5/132*(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)*((3^(1/2)*(n+39/11)*11^(1/2)-17/5*n-191/5)*(17+3*11^(1/2)*3^(1/2))^(1/3)+(3^(1/2)*(-17/10*n+747/55)*11^(1/2)+103/10*n-373/5)*(17+3*11^(1/2)*3^(1/2))^(2/3)+8/5*n+44/5))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(-330 \left(n +\frac{39}{11}\right) \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{11}+374 \left(n +\frac{191}{17}\right) \left(1+\mathrm{I} \sqrt{3}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-561 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{1494}{187}\right) \sqrt{11}+1133 \left(n -\frac{746}{103}\right) \left(\mathrm{I} \sqrt{3}-1\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+352 n +1936\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}}{5808}\\+\\\frac{\left(\left(330 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n +\frac{39}{11}\right) \sqrt{11}-374 \left(\mathrm{I} \sqrt{3}-1\right) \left(n +\frac{191}{17}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(561 \left(n -\frac{1494}{187}\right) \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{11}-1133 \left(n -\frac{746}{103}\right) \left(1+\mathrm{I} \sqrt{3}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+352 n +1936\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}}{5808}\\+\\\frac{5 \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} \left(\left(\sqrt{3}\, \left(n +\frac{39}{11}\right) \sqrt{11}-\frac{17 n}{5}-\frac{191}{5}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{3}\, \left(-\frac{17 n}{10}+\frac{747}{55}\right) \sqrt{11}+\frac{103 n}{10}-\frac{373}{5}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{8 n}{5}+\frac{44}{5}\right)}{132} & \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) = 50
a(6) = 111
a(7) = 245
a(8) = 525
a(9) = 1100
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), 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) = 50
a \! \left(6\right) = 111
a \! \left(7\right) = 245
a \! \left(8\right) = 525
a \! \left(9\right) = 1100
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}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16507/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[44,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[16,x]
F[31,x] = F[32,x]
F[32,x] = F[29,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[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,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[32,x]+F[7,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[51,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[28,x]+F[4,x]
F[51,x] = F[40,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[12,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \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_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \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_{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_{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_{32}\! \left(x \right)+F_{7}\! \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_{51}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{40}\! \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_{12}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_44(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_16(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_29(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))
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_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_32(x) + F_7(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_51(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_28(x) + F_4(x))
Eq(F_51(x), F_40(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_12(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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