0123_0231_1320_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 45, 80, 142, 257, 464, 841, 1532, 2799, 5126, 9403, 17266, 31725, 58316, 107221, 197168, 362603, 666882, 1226535, 2255894, 4149177, 7631464, 14036385, 25816868, 47484551, 87337630, 160638867, 295460858, 543437157, 999536676, 1838434477, 3381408088, 6219379011, 11439221338, 21040008191, 38698608286, 71177837553, 130916453760, 240792899321, 442887190348, 814596543135, 1498276632502, 2755760365675, 5068633540994, 9322670538845, 17147064445180, 31538368524677, 58008103508352, 106693536477851, 196240008510514, 360941648496343, 663875193484326, 1221056850490793, 2245873692471064, 4130805736445777, 7597736279407220, 13974415708323639, 25702957724176206, 47275109711906627, 86952483144406026, 159930550580488405, 294158143436800596, 541041177161694557, 995129871178983080, 1830329191777477747, 3366500240118154890, 6191959303074615215, 11388788734970247342, 20947248278163016929, 38527996316207878960, 70864033329341142697, 130339277923712038044, 239731307569261059151, 440934618822314239334, 811005204315287335963, 1491671130706862633874, 2743610953844464208589, 5046287288866614177836, 9281569373417941019701, 17071467616129019405520, 31399324278413574602443, 57752361267960535027042, 106223153162503129034375, 195374838708877238663222, 359350353139340902723993, 660948345010721270420936, 1215673536858939411807489, 2235972235009001584951748, 4112594116878662267179495, 7564239888746603263938046, 13912806240634267116068595, 25589640246259532647185434, 47066686375640403027191365, 86569132862534202790444676, 159225459484434138464820749, 292861278722608744282456056, 538655871069577085537720739
Generating function in Maple syntax:
(x^11+2*x^10-3*x^9-4*x^8+4*x^7+16*x^6-6*x^4-2*x^3-x^2+2*x-1)/(x^3+x^2+x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{11}+2 x^{10}-3 x^{9}-4 x^{8}+4 x^{7}+16 x^{6}-6 x^{4}-2 x^{3}-x^{2}+2 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**11 + 2*x**10 - 3*x**9 - 4*x**8 + 4*x**7 + 16*x**6 - 6*x**4 - 2*x**3 - x**2 + 2*x - 1)/((x - 1)**2*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3+x^2+x-1)*(x-1)^2*F(x)+x^11+2*x^10-3*x^9-4*x^8+4*x^7+16*x^6-6*x^4-2*x^3-x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{11}+2 x^{10}-3 x^{9}-4 x^{8}+4 x^{7}+16 x^{6}-6 x^{4}-2 x^{3}-x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
-17+4*n-9/22*Sum(_alpha^(1-n),_alpha = RootOf(_Z^3+_Z^2+_Z-1))+20/11*Sum(_alpha^(-n),_alpha = RootOf(_Z^3+_Z^2+_Z-1))+3/22*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3+_Z^2+_Z-1))+piecewise(n = 0,12,n = 1,13,n = 2,5,n = 3,-2,n = 5,3,n = 6,1,0)
Explicit closed form in latex syntax:
-17+4 n -\frac{9 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{22}+\frac{20 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{11}+\frac{3 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{22}+\left(\left\{\begin{array}{cc}12 & n =0 \\ 13 & n =1 \\ 5 & n =2 \\ -2 & n =3 \\ 3 & n =5 \\ 1 & 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) = 45
a(6) = 80
a(7) = 142
a(8) = 257
a(9) = 464
a(10) = 841
a(11) = 1532
a(n+3) = a(n)+a(n+1)+a(n+2)-8*n+34, 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) = 45
a \! \left(6\right) = 80
a \! \left(7\right) = 142
a \! \left(8\right) = 257
a \! \left(9\right) = 464
a \! \left(10\right) = 841
a \! \left(11\right) = 1532
a \! \left(n +3\right) = a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)-8 n +34, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17527/
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[18,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]+F[43,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[41,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[11,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[19,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[17,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[19,x]+F[38,x]
F[43,x] = F[24,x]+F[44,x]+F[64,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[54,x]
F[46,x] = F[47,x]+F[7,x]
F[47,x] = F[24,x]+F[48,x]+F[49,x]
F[48,x] = F[17,x]*F[4,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[11,x]+F[52,x]
F[52,x] = F[48,x]
F[53,x] = F[48,x]
F[54,x] = F[55,x]+F[59,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[19,x]+F[32,x]
F[59,x] = 2*F[24,x]+F[60,x]+F[62,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[32,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[38,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[77,x]
F[66,x] = F[19,x]+F[67,x]
F[67,x] = F[24,x]+F[41,x]+F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[11,x]+F[71,x]
F[71,x] = F[24,x]+F[48,x]+F[72,x]
F[72,x] = F[4,x]*F[51,x]
F[73,x] = F[32,x]+F[74,x]
F[74,x] = 2*F[24,x]+F[60,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[38,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[39,x]
F[79,x] = F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,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_{18}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{43}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \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_{19}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \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_{17}\! \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_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)+F_{64}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{7}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \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_{19}\! \left(x \right)+F_{32}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{38}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{77}\! \left(x \right)
F_{66}\! \left(x \right) = F_{19}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{11}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{73}\! \left(x \right) = F_{32}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{60}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{38}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{39}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \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_18(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_43(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_41(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_11(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_19(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_17(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_19(x) + F_38(x))
Eq(F_43(x), F_24(x) + F_44(x) + F_64(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_54(x))
Eq(F_46(x), F_47(x) + F_7(x))
Eq(F_47(x), F_24(x) + F_48(x) + F_49(x))
Eq(F_48(x), F_17(x)*F_4(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_11(x) + F_52(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_48(x))
Eq(F_54(x), F_55(x) + F_59(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_19(x) + F_32(x))
Eq(F_59(x), 2*F_24(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_32(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_38(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_77(x))
Eq(F_66(x), F_19(x) + F_67(x))
Eq(F_67(x), F_24(x) + F_41(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_11(x) + F_71(x))
Eq(F_71(x), F_24(x) + F_48(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_51(x))
Eq(F_73(x), F_32(x) + F_74(x))
Eq(F_74(x), 2*F_24(x) + F_60(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_38(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_39(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_52(x))
Pack JSON:
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Specification JSON:
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