0231_0321_2103_2130
Counting sequence:
1, 1, 2, 6, 20, 63, 183, 499, 1303, 3305, 8218, 20149, 48895, 117730, 281757, 671070, 1592091, 3765145, 8880745, 20900778, 49099356, 115163858, 269766692, 631217652, 1475575115, 3446639896, 8045153208, 18768086893, 43761257778, 101994079697, 237630463988, 553469797120, 1288753156250, 3000171564433, 6982920629357, 16250056817869, 37810270563661, 87965158292747, 204628069784520, 475970321762599, 1107032300872621, 2574608962598756, 5987381875256067, 13923258536327456, 32376217890836505, 75282712614644883, 175045348877656663, 406999615852650964, 946296415533551978, 2200146470005544740, 5115268407267248338, 11892653122254340654, 27649257872094335185, 64281113478762101730, 149444233213793032542, 347433241354831342215, 807719392713434287696, 1877789971839961658635, 4365472457838904636170, 10148772995332338664338, 23593596387723395948576, 54849580321472870290163, 127512050566561067909907, 296434046208974679640727, 689134535344302414476275, 1602061399239574234222941, 3724376843355203117686438, 8658197607162984427746633, 20128011214482563539401803, 46792236633405809395020182, 108779327800273054563244633, 252882406792205646497575250, 587882572296907715878214215, 1366665772881447061436649533, 3177121664392455703518918517, 7385930186810829312497314414, 17170241338727920809762040616, 39916041975352353943552978662, 92793686772207638395117300720, 215719444378237305850978245784, 501487454912831350192193632391, 1165818069448591788241622739036, 2710200556287079392135221260532, 6300456611654344794002136057681, 14646794045392385073752636631390, 34049683976265380400938955516853, 79155949463885386600413922297168, 184015332512957125575316878389204, 427783898644244666930462517886678, 994477212484462502249969262768325, 2311879636905370030481902718096233, 5374469312846459217911438320985809, 12494127734123159023090891138223625, 29045327927402046520072524188172255, 67522205054446516444228878003804852, 156970101897939086740195775562788635, 364911253222887216701328692447914201, 848315820348351878766076599379937400, 1972095175339464061032274579122702495, 4584565376229063185059012011688002548, 10657822073725588397464589037462246821
Generating function in Maple syntax:
-(3*x^6-14*x^5+25*x^4-29*x^3+20*x^2-7*x+1)/(2*x-1)/(x^3-2*x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{3 x^{6}-14 x^{5}+25 x^{4}-29 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-3*x**6 + 14*x**5 - 25*x**4 + 29*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**3*(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)+3*x^6-14*x^5+25*x^4-29*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{6}-14 x^{5}+25 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
1/13800*(230*2^(2/3)*((48/23*I-16/23*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200-713*2^(1/3)*((-147/713*I-49/713*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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*(-230*2^(2/3)*((48/23*I+16/23*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200+713*((-147/713*I+49/713*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-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*((320*2^(2/3)*23^(1/2)*3^(1/2)+460*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200+(-98*23^(1/2)*3^(1/2)*2^(1/3)+1426*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)-1/2*n^2+1/2*n-3*2^n+2
Explicit closed form in latex syntax:
\frac{\left(230 \,2^{\frac{2}{3}} \left(\left(\frac{48 \,\mathrm{I}}{23}-\frac{16 \sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200-713 \,2^{\frac{1}{3}} \left(\left(-\frac{147 \,\mathrm{I}}{713}-\frac{49 \sqrt{3}}{713}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{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(-230 \,2^{\frac{2}{3}} \left(\left(\frac{48 \,\mathrm{I}}{23}+\frac{16 \sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+713 \left(\left(-\frac{147 \,\mathrm{I}}{713}+\frac{49 \sqrt{3}}{713}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{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(\left(320 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}+460 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+\left(-98 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+1426 \,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}-\frac{n^{2}}{2}+\frac{n}{2}-3 \,2^{n}+2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 183
a(n+4) = -2*a(n)+5*a(n+1)-8*a(n+2)+5*a(n+3)-1/2*(n+2)*(n-3), 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 183
a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)-\frac{\left(n +2\right) \left(n -3\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15846/
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[17,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[14,x]+F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[35,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[11,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[14,x]+F[32,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[17,x]+F[37,x]
F[37,x] = F[38,x]+F[49,x]
F[38,x] = F[19,x]+F[39,x]+F[47,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[11,x]+F[42,x]
F[42,x] = F[19,x]+F[43,x]+F[45,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[41,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[11,x]+F[29,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[2,x]+F[38,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[14,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[52,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \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_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \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_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_14(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_35(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_11(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_14(x) + F_32(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_17(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_49(x))
Eq(F_38(x), F_19(x) + F_39(x) + F_47(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_11(x) + F_42(x))
Eq(F_42(x), F_19(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_41(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_11(x) + F_29(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_2(x) + F_38(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_14(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_52(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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