0231_0321_1203_2130
Counting sequence:
1, 1, 2, 6, 20, 62, 182, 521, 1468, 4086, 11262, 30797, 83679, 226168, 608606, 1631687, 4360932, 11624247, 30914306, 82054363, 217424003, 575274419, 1520154825, 4012518443, 10580931296, 27877966618, 73396308880, 193109129434, 507786598187, 1334564222920, 3505933875834, 9206549223544, 24167907198955, 63423156753129, 166394441210504, 436441025973140, 1144511442072851, 3000772876297776, 7866361163344453, 20618196196566720, 54034455797853529, 141592639122349035, 370993294185094573, 971968033801425161, 2546260981018205515, 6669953682529185242, 17470896829167773052, 45759699720016439326, 119847636324087609338, 313874882164378393941, 821990123834073764832, 2152590918501863531502, 5636934364745658343074, 14760889630294520968373, 38651958852831870578151, 101209456732238311386492, 265010049557166202881002, 693898891297729239930827, 1816868415788899286955476, 4757128969923947185018643, 12455500952000591362935898, 32611657823873694830173299, 85384782030827777281148895, 223555031384657402716035023, 585309006386669027594495541, 1532438693845035222998111323, 4012162147946491842439500500, 10504408250512759418305596314, 27501900665620318895284242432, 72003242004195211689037242750, 188512354496967613584669119279, 493544350483049024943321745924, 1292145173899046068476157855850, 3382948073212008507833875405484, 8856831326833345618488267268299, 23187853423528154790552737692145, 60707443832276685082358505454796, 158936139987494712721707633934456, 416104839611974883662863654134623, 1089387360353872171805999950949800, 2852078120916626283477472405846049, 7466895541267843597295291900807440, 19548721342073888194509471407451393, 51179530804238082858920913592865191, 133990480888991015546783381597568385, 350793329518405389215745053447350797, 918392803315819379947026812305486547, 2404392741884845238460958951883519006, 6294803233062575768683886503713806708, 16480058362212469619510137837096292046, 43145468108311669366944389488767179902, 112956569727837731642025779977473533457, 295724761265983020038964910803912447844, 774218923366962261870057197299696431078, 2026934820109008769957810244319224939446, 5306592072379458688852639695959516454797, 13892856590036192143769888442881275060447, 36372013017184908006654435673271132592032, 95223264569291647615295355660350231108646, 249297971568104626375312402076182188817423, 652671093871175564904877698747154826673668
Generating function in Maple syntax:
-(x^7-x^6-2*x^5+3*x^4-4*x^3+8*x^2-5*x+1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
-\frac{x^{7}-x^{6}-2 x^{5}+3 x^{4}-4 x^{3}+8 x^{2}-5 x +1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(-x**7 + x**6 + 2*x**5 - 3*x**4 + 4*x**3 - 8*x**2 + 5*x - 1)/((x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)+x^7-x^6-2*x^5+3*x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{7}-x^{6}-2 x^{5}+3 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
564/115*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))-2499/115*Sum(_alpha^(2-n),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))+4474/115*Sum(_alpha^(1-n),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))-4492/115*Sum(_alpha^(-n),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))+1236/115*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))-piecewise(n = 0,8,n = 1,4,n = 2,1,0)
Explicit closed form in latex syntax:
\frac{564 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{115}-\frac{2499 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{115}+\frac{4474 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{115}-\frac{4492 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{115}+\frac{1236 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{115}-\left(\left\{\begin{array}{cc}8 & n =0 \\ 4 & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 182
a(7) = 521
a(n+5) = a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*a(n+4), n >= 8
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) = 62
a \! \left(6\right) = 182
a \! \left(7\right) = 521
a \! \left(n +5\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15811/
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[10,x]*F[4,x]
F[4,x] = F[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[13,x]+F[14,x]
F[12,x] = 0
F[13,x] = F[10,x]*F[6,x]
F[14,x] = F[10,x]*F[15,x]
F[15,x] = F[9,x]
F[16,x] = F[17,x]+F[38,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]*F[19,x]
F[19,x] = F[20,x]+F[28,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[23,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[10,x]*F[27,x]
F[27,x] = F[24,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[10,x]*F[31,x]
F[31,x] = F[28,x]+F[32,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[34,x]
F[34,x] = F[10,x]*F[17,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[32,x]
F[38,x] = F[12,x]+F[39,x]+F[53,x]
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[41,x]+F[47,x]
F[41,x] = F[42,x]+F[6,x]
F[42,x] = F[43,x]
F[43,x] = F[10,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[10,x]+F[46,x]
F[46,x] = F[43,x]
F[47,x] = F[38,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[10,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[33,x]+F[52,x]
F[52,x] = F[49,x]
F[53,x] = F[10,x]*F[54,x]
F[54,x] = F[16,x]+F[55,x]
F[55,x] = F[56,x]+F[65,x]
F[56,x] = F[12,x]+F[34,x]+F[57,x]
F[57,x] = F[10,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[10,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = x^2
F[62,x] = F[33,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[10,x]*F[33,x]
F[65,x] = F[12,x]+F[66,x]+F[70,x]+F[71,x]
F[66,x] = F[10,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[46,x]
F[69,x] = F[52,x]
F[70,x] = F[10,x]*F[38,x]
F[71,x] = F[10,x]*F[72,x]
F[72,x] = F[55,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{16}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{38}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{53}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{6}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)
F_{47}\! \left(x \right) = F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right)+F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = x^{2}
F_{62}\! \left(x \right) = F_{33}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{46}\! \left(x \right)
F_{69}\! \left(x \right) = F_{52}\! \left(x \right)
F_{70}\! \left(x \right) = F_{10}\! \left(x \right) F_{38}\! \left(x \right)
F_{71}\! \left(x \right) = F_{10}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{55}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_16(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x) + F_14(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_10(x)*F_6(x))
Eq(F_14(x), F_10(x)*F_15(x))
Eq(F_15(x), F_9(x))
Eq(F_16(x), F_17(x) + F_38(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_28(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_23(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_10(x)*F_27(x))
Eq(F_27(x), F_24(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_10(x)*F_31(x))
Eq(F_31(x), F_28(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_10(x)*F_17(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_32(x))
Eq(F_38(x), F_12(x) + F_39(x) + F_53(x))
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_47(x))
Eq(F_41(x), F_42(x) + F_6(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_10(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_10(x) + F_46(x))
Eq(F_46(x), F_43(x))
Eq(F_47(x), F_38(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_10(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_33(x) + F_52(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_10(x)*F_54(x))
Eq(F_54(x), F_16(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_65(x))
Eq(F_56(x), F_12(x) + F_34(x) + F_57(x))
Eq(F_57(x), F_10(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_10(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), x**2)
Eq(F_62(x), F_33(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_10(x)*F_33(x))
Eq(F_65(x), F_12(x) + F_66(x) + F_70(x) + F_71(x))
Eq(F_66(x), F_10(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_46(x))
Eq(F_69(x), F_52(x))
Eq(F_70(x), F_10(x)*F_38(x))
Eq(F_71(x), F_10(x)*F_72(x))
Eq(F_72(x), F_55(x))
Pack JSON:
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Specification JSON:
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