0123_0231_1320_2013_3102
Counting sequence:
1, 1, 2, 6, 19, 47, 99, 192, 356, 646, 1163, 2092, 3773, 6831, 12415, 22638, 41386, 75806, 139043, 255274, 468967, 861911, 1584539, 2913540, 5357824, 9853422, 18121963, 33330016, 61301809, 112749767, 207377111, 381423714, 701545094, 1290339862, 2373302019, 4365179694, 8028813627, 14767286687, 27161270611, 49957360744, 91885907036, 169004526518, 310847781515, 571738201332, 1051590494629, 1934176461695, 3557505140783, 6543272079094, 12034953662370, 22135730861806, 40713956581539, 74884641082642, 137734328501519, 253332926139783, 465951895696523, 857019150308844, 1576303972114552, 2899275018087646, 5332598140477035, 9808177130643432, 18040050289170457, 33180825560251351, 61029052980023687, 112249928829401898, 206459807369631230, 379738789179008934, 698448525377991939, 1284647121926579670, 2362834436483525731, 4345930083788040079, 7993411642198085699, 14702176162469589136, 27041517888455649876, 49737105693123256934, 91480799744048425355, 168259423325627258684, 309477328762798864525, 569217551832474469071, 1046954303920900509663, 1925649184516173757438, 3541821040269548647066, 6514424528706622821694, 11981894753492345130275, 22038140322468516499578, 40534459604667484348471, 74554494680628345871543, 137127094607764346609019, 252216048893060176714580, 463897638181452869076720, 853240781682277392277838, 1569354468756790437942507, 2886492888620520699166192, 5309088139059588529251329, 9764935496436899666220391, 17960516524117008894493751, 33034540159613497089816690, 60759992180167405650377334, 111755048863897911634529462, 205549581203678814374560259, 378064622247744131659298814, 695369252315320857668215179
Generating function in Maple syntax:
-(x^9+2*x^8+x^7-x^6-8*x^5+3*x^4-2*x^3+6*x^2-4*x+1)/(x^3+x^2+x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{x^{9}+2 x^{8}+x^{7}-x^{6}-8 x^{5}+3 x^{4}-2 x^{3}+6 x^{2}-4 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**9 - 2*x**8 - x**7 + x**6 + 8*x**5 - 3*x**4 + 2*x**3 - 6*x**2 + 4*x - 1)/((x - 1)**4*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x-1)^4*F(x)+x^9+2*x^8+x^7-x^6-8*x^5+3*x^4-2*x^3+6*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{9}+2 x^{8}+x^{7}-x^{6}-8 x^{5}+3 x^{4}-2 x^{3}+6 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/528*(((-39*11^(1/2)+165*I)*3^(1/2)-117*I*11^(1/2)+165)*(17+3*11^(1/2)*3^(1/2))^(1/3)+396+((42*11^(1/2)+264*I)*3^(1/2)-126*I*11^(1/2)-264)*(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)+1/528*(((42*11^(1/2)-264*I)*3^(1/2)+126*I*11^(1/2)-264)*(17+3*11^(1/2)*3^(1/2))^(2/3)+396+((-39*11^(1/2)-165*I)*3^(1/2)+117*I*11^(1/2)+165)*(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)+1/528*((-84*11^(1/2)*3^(1/2)+528)*(17+3*11^(1/2)*3^(1/2))^(2/3)+78*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-330*(17+3*11^(1/2)*3^(1/2))^(1/3)+396)*(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)+1/12*n^3+5/4*n^2-34/3*n+47/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(-39 \sqrt{11}+165 \,\mathrm{I}\right) \sqrt{3}-117 \,\mathrm{I} \sqrt{11}+165\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+396+\left(\left(42 \sqrt{11}+264 \,\mathrm{I}\right) \sqrt{3}-126 \,\mathrm{I} \sqrt{11}-264\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{528}\\+\\\frac{\left(\left(\left(42 \sqrt{11}-264 \,\mathrm{I}\right) \sqrt{3}+126 \,\mathrm{I} \sqrt{11}-264\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+396+\left(\left(-39 \sqrt{11}-165 \,\mathrm{I}\right) \sqrt{3}+117 \,\mathrm{I} \sqrt{11}+165\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\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}}{528}\\+\\\frac{\left(\left(-84 \sqrt{11}\, \sqrt{3}+528\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+78 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-330 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+396\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}}{528}\\+\frac{n^{3}}{12}+\frac{5 n^{2}}{4}-\frac{34 n}{3}+\frac{47}{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) = 47
a(6) = 99
a(7) = 192
a(8) = 356
a(9) = 646
a(n+3) = -1/6*n^3-5/2*n^2+a(n)+a(n+1)+a(n+2)+71/3*n-17, 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) = 47
a \! \left(6\right) = 99
a \! \left(7\right) = 192
a \! \left(8\right) = 356
a \! \left(9\right) = 646
a \! \left(n +3\right) = -\frac{n^{3}}{6}-\frac{5 n^{2}}{2}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+\frac{71 n}{3}-17, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17511/
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[23,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[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[11,x]
F[22,x] = F[18,x]
F[23,x] = F[24,x]+F[36,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[1,x]+F[14,x]
F[28,x] = F[24,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[24,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[24,x]
F[36,x] = F[37,x]+F[38,x]+F[45,x]
F[37,x] = 0
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[41,x]+F[7,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[17,x]
F[44,x] = F[36,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[60,x]
F[47,x] = F[24,x]+F[48,x]
F[48,x] = F[37,x]+F[49,x]+F[55,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[11,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[17,x]*F[4,x]
F[54,x] = F[48,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[24,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[11,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,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_{23}\! \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_{18}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{36}\! \left(x \right)
F_{24}\! \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_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \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_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{24}\! \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_{24}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{37}\! \left(x \right) = 0
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_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{7}\! \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_{17}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{48}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \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_23(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_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_11(x))
Eq(F_22(x), F_18(x))
Eq(F_23(x), F_24(x) + F_36(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_1(x) + F_14(x))
Eq(F_28(x), F_24(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_24(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_24(x))
Eq(F_36(x), F_37(x) + F_38(x) + F_45(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_41(x) + F_7(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_17(x))
Eq(F_44(x), F_36(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_60(x))
Eq(F_47(x), F_24(x) + F_48(x))
Eq(F_48(x), F_37(x) + F_49(x) + F_55(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_11(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_17(x)*F_4(x))
Eq(F_54(x), F_48(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_24(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_11(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(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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