0132_1230_1320_2103_3021
Counting sequence:
1, 1, 2, 6, 19, 48, 101, 196, 367, 676, 1237, 2260, 4131, 7560, 13853, 25412, 46655, 85708, 157517, 289572, 532435, 979104, 1800629, 3311620, 6090735, 11202292, 20603877, 37896052, 69701283, 128200184, 235796397, 433696644, 797691903, 1467183516, 2698570525, 4963444292, 9129196563, 16791209488, 30883848325, 56804252228, 104479307759, 192167405892, 353450963317, 650097674260, 1195716040611, 2199264675176, 4045078386877, 7440059099332, 13684402157887, 25169539640428, 46294000893805, 85147942688100, 156611483218131, 288053426795648, 529812852697301, 974477762706308, 1792344042194287, 3296634657592724, 6063456462487941, 11152435162269364, 20512526282344227, 37728417907095512, 69393379351702861, 127634323541136132, 234756120799927807, 431783823692759868, 794174268033816637, 1460714212526496900, 2686672304253065747, 4941560784813371376, 9088947301592925861, 16717180390659354564, 30747688477065643119, 56553816169317914596, 104018685037042903061, 191320189683426451284, 351892690889787259171, 647231565610256603464, 1190444446183470303581, 2189568702683514155588, 4027244714477241051711, 7407257863344225499660, 13624071280504980695437, 25058573858326447234980, 46089903002175653417939, 84772548141007081335904, 155921025001509181976053, 286783476144691916716804, 527477049287208180015343, 970181550433409278694452, 1784442075865309375412517, 3282100675585926834107892, 6036724301884645488200099, 11103267053335881697705400, 20422092030806454019997933, 37562083386026981205887620, 69087442470169316923574783, 127071617887002752149443804, 233721143743199050278889309, 429880204100371119351890628, 790672965730572921780206099
Generating function in Maple syntax:
(2*x^9+x^8-3*x^7-7*x^6-3*x^5+4*x^4+x^3+3*x^2-3*x+1)/(x^3+x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{2 x^{9}+x^{8}-3 x^{7}-7 x^{6}-3 x^{5}+4 x^{4}+x^{3}+3 x^{2}-3 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x**9 + x**8 - 3*x**7 - 7*x**6 - 3*x**5 + 4*x**4 + x**3 + 3*x**2 - 3*x + 1)/((x - 1)**3*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3+x^2+x-1)*(x-1)^3*F(x)+2*x^9+x^8-3*x^7-7*x^6-3*x^5+4*x^4+x^3+3*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{9}+x^{8}-3 x^{7}-7 x^{6}-3 x^{5}+4 x^{4}+x^{3}+3 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,n = 3,6,1/88*(((22*I-6*11^(1/2))*3^(1/2)-18*I*11^(1/2)+22)*(17+3*11^(1/2)*3^(1/2))^(1/3)+88+((55*I+9*11^(1/2))*3^(1/2)-27*I*11^(1/2)-55)*(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)+1/88*(((9*11^(1/2)-55*I)*3^(1/2)+27*I*11^(1/2)-55)*(17+3*11^(1/2)*3^(1/2))^(2/3)+88+((-6*11^(1/2)-22*I)*3^(1/2)+18*I*11^(1/2)+22)*(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)+1/88*((-18*11^(1/2)*3^(1/2)+110)*(17+3*11^(1/2)*3^(1/2))^(2/3)+12*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)-44*(17+3*11^(1/2)*3^(1/2))^(1/3)+88)*(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)+n^2-6*n-4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(\left(22 \,\mathrm{I}-6 \sqrt{11}\right) \sqrt{3}-18 \,\mathrm{I} \sqrt{11}+22\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+88+\left(\left(55 \,\mathrm{I}+9 \sqrt{11}\right) \sqrt{3}-27 \,\mathrm{I} \sqrt{11}-55\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}}{88}\\+\\\frac{\left(\left(\left(9 \sqrt{11}-55 \,\mathrm{I}\right) \sqrt{3}+27 \,\mathrm{I} \sqrt{11}-55\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+88+\left(\left(-6 \sqrt{11}-22 \,\mathrm{I}\right) \sqrt{3}+18 \,\mathrm{I} \sqrt{11}+22\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}}{88}\\+\\\frac{\left(\left(-18 \sqrt{11}\, \sqrt{3}+110\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+12 \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}-44 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+88\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}}{88}\\+n^{2}-6 n -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) = 48
a(6) = 101
a(7) = 196
a(8) = 367
a(9) = 676
a(n+3) = -2*n^2+a(n)+a(n+1)+a(n+2)+12*n+12, 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) = 48
a \! \left(6\right) = 101
a \! \left(7\right) = 196
a \! \left(8\right) = 367
a \! \left(9\right) = 676
a \! \left(n +3\right) = -2 n^{2}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+12 n +12, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/20081/
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[29,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[24,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[25,x]*F[4,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[18,x]
F[29,x] = F[30,x]+F[7,x]
F[30,x] = F[15,x]+F[31,x]+F[51,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[44,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[15,x]+F[20,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[11,x]+F[19,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[20,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[19,x]+F[43,x]
F[43,x] = F[41,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[23,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[26,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[25,x]+F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[27,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]+F[61,x]
F[59,x] = F[15,x]+F[20,x]+F[60,x]
F[60,x] = F[37,x]*F[4,x]
F[61,x] = F[41,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_{29}\! \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_{24}\! \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_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{51}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{44}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \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_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \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_{19}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \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_{23}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{26}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{27}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{61}\! \left(x \right) = F_{41}\! \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_29(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_24(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_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_18(x))
Eq(F_29(x), F_30(x) + F_7(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_51(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_44(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_15(x) + F_20(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_11(x) + F_19(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_20(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_19(x) + F_43(x))
Eq(F_43(x), F_41(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_23(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_26(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_25(x) + F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_27(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x) + F_61(x))
Eq(F_59(x), F_15(x) + F_20(x) + F_60(x))
Eq(F_60(x), F_37(x)*F_4(x))
Eq(F_61(x), F_41(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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