0132_0321_1023_1230_2031
Counting sequence:
1, 1, 2, 6, 19, 50, 125, 311, 762, 1841, 4408, 10483, 24803, 58457, 137362, 322024, 753574, 1760968, 4110530, 9586679, 22343078, 52045740, 121183661, 282070966, 656384953, 1527104528, 3552284606, 8262101254, 19214488519, 44681938597, 103898051929, 241579919417, 561690248159, 1305927876968, 3036201780873, 7058842138277, 16410790677754, 38152250513587, 88696515548672, 206200440445445, 479369009102575, 1114418237689495, 2590745783531614, 6022822576615246, 14001491316715646, 32549752837211354, 75669426320121770, 175910867640968937, 408944612280252362, 950683567942933162, 2210076099171944305, 5137812675397269476, 11943982089604576289, 27766420364754588658, 64549152533316111142, 150058677944285138416, 348844294405646870347, 810964947063461168363, 1885265421784060120565, 4382711569568033924807, 10188570474799031671911, 23685556765042490504186, 55062246539631563310793, 128004206908739854000927, 297573703439661563725142, 691774978038464418946385, 1608181798514931607096588, 3738569261302639681130707, 8691119382679081878322711, 20204401824474693267696049, 46969536706054024879652126, 109190927206867596152035808, 253838112525158522674619474, 590102024451644786631786696, 1371820783942500213610773782, 3189096430956424369898452279, 7413749777957837937705645418, 17234877308363155838098556612, 40066228896619181808205716713, 93142682028559948505316368350, 216530465927224470907417550821, 503372263221552373308220526512, 1170198540943038512392509748882, 2720381564344305156204048863702, 6324119878104110773436652535787, 14701795053437820757108584566669, 34177526981087395968507560465157, 79453110737735287445708747624705, 184706073347222431762851567248107, 429389525625920137340094273497512, 998209541065685027801336383722893, 2320555645558460993927801488798709, 5394637380659227029797452746464110, 12541010392826396040916895687012875, 29154312064374464688380143189797388, 67775552791174344522256081890230125, 157559044643200237282952210100891419, 366280340421926009685200515916082215, 851498484789495673765608461711947298, 1979493818202678273207535077006676014, 4601764825515055812720878495593646978
Generating function in Maple syntax:
-(x^9-2*x^8-3*x^7+x^6+3*x^5-4*x^4-x^3-2*x^2+3*x-1)/(x^3-2*x^2+3*x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{x^{9}-2 x^{8}-3 x^{7}+x^{6}+3 x^{5}-4 x^{4}-x^{3}-2 x^{2}+3 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**9 + 2*x**8 + 3*x**7 - x**6 - 3*x**5 + 4*x**4 + x**3 + 2*x**2 - 3*x + 1)/((x**3 - 2*x**2 + 3*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x^3+x^2+x-1)*F(x)+x^9-2*x^8-3*x^7+x^6+3*x^5-4*x^4-x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{9}-2 x^{8}-3 x^{7}+x^{6}+3 x^{5}-4 x^{4}-x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-7709/5566*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+3853/5566*Sum(_alpha^(3-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-16109/5566*Sum(_alpha^(2-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-3002/2783*Sum(_alpha^(1-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-33801/5566*Sum(_alpha^(-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+9728/2783*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+piecewise(n = 0,2,n = 1,6,n = 2,1,n = 3,-1,0)
Explicit closed form in latex syntax:
-\frac{7709 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{5566}+\frac{3853 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{5566}-\frac{16109 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{5566}-\frac{3002 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{2783}-\frac{33801 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{5566}+\frac{9728 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2783}+\left(\left\{\begin{array}{cc}2 & n =0 \\ 6 & n =1 \\ 1 & n =2 \\ -1 & n =3 \\ 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) = 19
a(5) = 50
a(6) = 125
a(7) = 311
a(8) = 762
a(9) = 1841
a(n+6) = -a(n)+a(n+1)-2*a(n+2)+a(n+3)-4*a(n+4)+4*a(n+5), 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) = 125
a \! \left(7\right) = 311
a \! \left(8\right) = 762
a \! \left(9\right) = 1841
a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19309/
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[22,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[21,x]
F[21,x] = F[16,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[15,x]+F[24,x]+F[60,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[37,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[1,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[17,x]+F[33,x]
F[33,x] = F[18,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[18,x]
F[37,x] = F[38,x]+F[59,x]
F[38,x] = F[15,x]+F[24,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[15,x]+F[42,x]+F[58,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[4,x]+F[45,x]
F[45,x] = F[31,x]
F[46,x] = F[41,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[56,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[44,x]+F[54,x]
F[54,x] = F[51,x]+F[55,x]
F[55,x] = F[48,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[50,x]
F[58,x] = F[2,x]*F[4,x]
F[59,x] = F[48,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[64,x]
F[62,x] = F[2,x]+F[63,x]
F[63,x] = F[52,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = F[15,x]+F[42,x]+F[66,x]
F[66,x] = F[4,x]*F[62,x]
F[67,x] = F[57,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_{22}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{60}\! \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_{37}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \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_{18}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{59}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{58}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \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_{56}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{48}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{48}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{2}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{52}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = F_{57}\! \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_22(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_21(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_15(x) + F_24(x) + F_60(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_37(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_1(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_17(x) + F_33(x))
Eq(F_33(x), F_18(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_18(x))
Eq(F_37(x), F_38(x) + F_59(x))
Eq(F_38(x), F_15(x) + F_24(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_58(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_4(x) + F_45(x))
Eq(F_45(x), F_31(x))
Eq(F_46(x), F_41(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) + F_56(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_44(x) + F_54(x))
Eq(F_54(x), F_51(x) + F_55(x))
Eq(F_55(x), F_48(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_50(x))
Eq(F_58(x), F_2(x)*F_4(x))
Eq(F_59(x), F_48(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_64(x))
Eq(F_62(x), F_2(x) + F_63(x))
Eq(F_63(x), F_52(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), F_15(x) + F_42(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_62(x))
Eq(F_67(x), F_57(x))
Pack JSON:
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Specification JSON:
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