0123_0132_1320_2103_3102
Counting sequence:
1, 1, 2, 6, 19, 47, 89, 166, 317, 602, 1138, 2143, 4022, 7528, 14057, 26196, 48734, 90529, 167954, 311254, 576269, 1066046, 1970670, 3640655, 6722142, 12405908, 22885917, 42203620, 77802310, 143388365, 264197678, 486688254, 896377581, 1650666698, 3039239002, 5595192935, 10299514758, 18957272472, 34889722065, 64207576972, 118153381086, 217410557377, 400030202266, 736012704814, 1354130715373, 2491269437454, 4583185923558, 8431454957303, 15510552265150, 28532703973764, 52486863970805, 96549783812060, 177601168133558, 326689295895693, 600923544197510, 1105348784562230, 2033179697347101, 3739804875133978, 6878904278762114, 12652812621989135, 23273016468349974, 42807151832311912, 78736894078326457, 144823393997874468, 266377684683074398, 489954549152723009, 901182449001681122, 1657558080398935462, 3048765297282805773, 5607619442974345470, 10314126655676475998, 18970808847244939647, 34893036232227462814, 64178750472791892564, 118043855576239010829, 217117681042876096116, 399343585877499445222, 734510460043824727789, 1350980363297002890462, 2484848383098339860430, 4570361816651982896973, 8406227147140153863114, 15461496541196120254058, 28438181283386728862935, 52306059944427118462438, 96205988520112554910552, 176950635471733105049377, 325463340411182068566940, 598621026601743721484894, 1101036721158284178203809, 2025123869043551244316266, 3724786116349545703168190, 6850953986969688960912109, 12600875752327060302783630, 23176634916028877196474838, 42628495495672483084168551, 78406056064757859437035902, 144211267217535515195286148, 265245949419771592047706341, 487863484084946996488850988, 897321042746941867871881814
Generating function in Maple syntax:
(x+1)*(3*x^10+5*x^9-x^8-12*x^7-14*x^6+x^5+7*x^4+x^3+x^2-2*x+1)/(x^2+x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{\left(x +1\right) \left(3 x^{10}+5 x^{9}-x^{8}-12 x^{7}-14 x^{6}+x^{5}+7 x^{4}+x^{3}+x^{2}-2 x +1\right)}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x + 1)*(3*x**10 + 5*x**9 - x**8 - 12*x**7 - 14*x**6 + x**5 + 7*x**4 + x**3 + x**2 - 2*x + 1)/((x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^3+x^2+x-1)*F(x)+(x+1)*(3*x^10+5*x^9-x^8-12*x^7-14*x^6+x^5+7*x^4+x^3+x^2-2*x+1) = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x +1\right) \left(3 x^{10}+5 x^{9}-x^{8}-12 x^{7}-14 x^{6}+x^{5}+7 x^{4}+x^{3}+x^{2}-2 x +1\right) = 0
Explicit closed form in Maple syntax:
-273/55*Sum(_alpha^(3-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-772/55*Sum(_alpha^(2-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-762/55*Sum(_alpha^(1-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-191/55*Sum(_alpha^(-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))+569/55*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-piecewise(n = 0,-3,Or(n = 1,n = 2),3,n = 3,6,n = 4,3,n = 5,-2,n = 6,-3,0)
Explicit closed form in latex syntax:
-\frac{273 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{55}-\frac{772 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{55}-\frac{762 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{55}-\frac{191 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{55}+\frac{569 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{55}-\left(\left\{\begin{array}{cc}-3 & n =0 \\ 3 & n =1\text{ or } n =2 \\ 6 & n =3 \\ 3 & n =4 \\ -2 & n =5 \\ -3 & n =6 \\ 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) = 47
a(6) = 89
a(7) = 166
a(8) = 317
a(9) = 602
a(10) = 1138
a(11) = 2143
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4), n >= 12
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) = 89
a \! \left(7\right) = 166
a \! \left(8\right) = 317
a \! \left(9\right) = 602
a \! \left(10\right) = 1138
a \! \left(11\right) = 2143
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16621/
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[24,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[16,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[19,x]+F[37,x]+F[56,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[19,x]+F[20,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[22,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[44,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[33,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[34,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[65,x]
F[58,x] = F[25,x]+F[59,x]
F[59,x] = F[19,x]+F[35,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[18,x]+F[4,x]
F[63,x] = F[34,x]+F[64,x]
F[64,x] = F[54,x]
F[65,x] = F[66,x]+F[70,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]+F[44,x]
F[70,x] = 2*F[19,x]+F[71,x]+F[74,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[18,x]+F[47,x]
F[74,x] = F[4,x]*F[66,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_{24}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \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_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \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_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{56}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \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_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{33}\! \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_{34}\! \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_{65}\! \left(x \right)
F_{58}\! \left(x \right) = F_{25}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{60}\! \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_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{63}\! \left(x \right) = F_{34}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{54}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{70}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right)+F_{44}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \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_24(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_16(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_12(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_56(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_19(x) + F_20(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_22(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_44(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_33(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_34(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_65(x))
Eq(F_58(x), F_25(x) + F_59(x))
Eq(F_59(x), F_19(x) + F_35(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_18(x) + F_4(x))
Eq(F_63(x), F_34(x) + F_64(x))
Eq(F_64(x), F_54(x))
Eq(F_65(x), F_66(x) + F_70(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x) + F_44(x))
Eq(F_70(x), 2*F_19(x) + F_71(x) + F_74(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_18(x) + F_47(x))
Eq(F_74(x), F_4(x)*F_66(x))
Pack JSON:
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Specification JSON:
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