0213_1032_1230_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 131, 323, 798, 1993, 5039, 12876, 33167, 85919, 223422, 582429, 1520731, 3974660, 10394955, 27196747, 71173470, 186288321, 487634263, 1276521820, 3341781239, 8748579207, 22903563646, 59961476213, 156979836643, 410976369748, 1075946580179, 2816859014291, 7374623413662, 19306999821049, 50546357594687, 132332043102444, 346449723397151, 907017048912815, 2374601296849470, 6216786636967437, 16275758282892715, 42610487675882276, 111555703877765403, 292056622554596635, 764614161516218334, 2001785858321434737, 5240743407505655911, 13720444354580479228, 35920589640678297863, 94041324542281876503, 246203383945437309694, 644568827228127492581, 1687503097632312585907, 4417940465496275122804, 11566318298577345057827, 30281014429784057183459, 79276724990043955900446, 207549160539165237058345, 543370756625538311222735, 1422563109334353679098252, 3724318571372513264508335, 9750392604775080635351231, 25526859242939613700905918, 66830185124022540047651325, 174963696129093671081693179, 458060903263202917417357892, 1199219013660425190029955051, 3139596137717927205752011243, 8219569399493121089165156958, 21519112060761055276762041633, 56337766782789428618078627959, 147494188287606233669450083996, 386144798080027659359205525527, 1010940205952474134469076635559, 2646675819777390521077868425342, 6929087253379690595855282827349, 18140585940361670210608578287491, 47492670567705302147181804452500, 124337425762754207286268785717875, 325519606720557272878167855766067, 852221394398917535570110035292830, 2231144576476195211220580806889561, 5841212335029667899701926195865183, 15292492428612808166883910147972140, 40036264950808756081558810425806399, 104816302423813459237400239674468047, 274412642320631620270858633320373566, 718421624538081399375000103554449133, 1880852231293612574294182845333545803, 4924135069342756317747414043704556388, 12891552976734656369627966065029063099, 33750523860861212776056256541889940411, 88360018605848981934140482730397005022, 231329531956685732986884643209564628689, 605628577264208216962632577628316681607, 1585556199835938917797651671965668770364, 4151040022243608536263080151288992783911, 10867563866894886690720985077211896089655, 28451651578441051535462029088677585147390, 74487390868428267914956652492462335522757, 195010521026843752208261632700681787253075
Generating function in Maple syntax:
(x^9-4*x^7+2*x^6+4*x^5+3*x^4+2*x^3-8*x^2+5*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{9}-4 x^{7}+2 x^{6}+4 x^{5}+3 x^{4}+2 x^{3}-8 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**9 - 4*x**7 + 2*x**6 + 4*x**5 + 3*x**4 + 2*x**3 - 8*x**2 + 5*x - 1)/((x - 1)**2*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x^2+x-1)*(x-1)^2*F(x)+x^9-4*x^7+2*x^6+4*x^5+3*x^4+2*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^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}-4 x^{7}+2 x^{6}+4 x^{5}+3 x^{4}+2 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,26,n = 1,10,n = 2,4,n = 3,1,0)-4*n-32/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+7/10*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+32/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-7/10*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-14*(3/2+1/2*5^(1/2))^(-n)+3/2*(1/2*5^(1/2)-1/2)^(-n)-14*(3/2-1/2*5^(1/2))^(-n)+3/2*(-1/2*5^(1/2)-1/2)^(-n)
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}26 & n =0 \\ 10 & n =1 \\ 4 & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-4 n -\frac{32 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{7 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{10}+\frac{32 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-\frac{7 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{10}-14 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\frac{3 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}-14 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\frac{3 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 52
a(6) = 131
a(7) = 323
a(8) = 798
a(9) = 1993
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)-4*n, 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) = 52
a \! \left(6\right) = 131
a \! \left(7\right) = 323
a \! \left(8\right) = 798
a \! \left(9\right) = 1993
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-4 n, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20559/
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[16,x]*F[4,x]
F[4,x] = F[24,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[10,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[9,x]
F[24,x] = F[25,x]+F[43,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[25,x]+F[29,x]
F[29,x] = F[30,x]+F[31,x]+F[37,x]
F[30,x] = 0
F[31,x] = F[16,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[16,x]
F[34,x] = F[35,x]
F[35,x] = F[30,x]+F[31,x]+F[36,x]
F[36,x] = F[16,x]*F[25,x]
F[37,x] = F[16,x]*F[38,x]
F[38,x] = F[25,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[16,x]*F[41,x]
F[41,x] = F[33,x]+F[42,x]
F[42,x] = F[40,x]
F[43,x] = F[30,x]+F[44,x]+F[67,x]
F[44,x] = F[16,x]*F[45,x]
F[45,x] = F[46,x]+F[53,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]
F[48,x] = F[16,x]*F[49,x]
F[49,x] = F[1,x]+F[47,x]
F[50,x] = F[51,x]
F[51,x] = F[16,x]*F[52,x]
F[52,x] = F[47,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[30,x]+F[44,x]+F[55,x]
F[55,x] = F[16,x]*F[56,x]
F[56,x] = F[25,x]+F[54,x]
F[57,x] = 2*F[30,x]+F[58,x]+F[65,x]
F[58,x] = F[16,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[61,x]
F[61,x] = F[16,x]*F[47,x]
F[62,x] = F[63,x]
F[63,x] = 2*F[30,x]+F[58,x]+F[64,x]
F[64,x] = F[16,x]*F[54,x]
F[65,x] = F[16,x]*F[66,x]
F[66,x] = F[54,x]
F[67,x] = F[16,x]*F[68,x]
F[68,x] = F[24,x]+F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[40,x]
F[71,x] = F[72,x]
F[72,x] = F[16,x]*F[73,x]
F[73,x] = F[69,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{24}\! \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_{16}\! \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_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{43}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{25}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{30}\! \left(x \right)+F_{44}\! \left(x \right)+F_{67}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{1}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{16}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{30}\! \left(x \right)+F_{44}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{58}\! \left(x \right) = F_{16}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{16}\! \left(x \right) F_{47}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{16}\! \left(x \right) F_{54}\! \left(x \right)
F_{65}\! \left(x \right) = F_{16}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{54}\! \left(x \right)
F_{67}\! \left(x \right) = F_{16}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{24}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{40}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{16}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_24(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_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_10(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_9(x))
Eq(F_24(x), F_25(x) + F_43(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_25(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_37(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_16(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_16(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_30(x) + F_31(x) + F_36(x))
Eq(F_36(x), F_16(x)*F_25(x))
Eq(F_37(x), F_16(x)*F_38(x))
Eq(F_38(x), F_25(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_16(x)*F_41(x))
Eq(F_41(x), F_33(x) + F_42(x))
Eq(F_42(x), F_40(x))
Eq(F_43(x), F_30(x) + F_44(x) + F_67(x))
Eq(F_44(x), F_16(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_53(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_16(x)*F_49(x))
Eq(F_49(x), F_1(x) + F_47(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_16(x)*F_52(x))
Eq(F_52(x), F_47(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_30(x) + F_44(x) + F_55(x))
Eq(F_55(x), F_16(x)*F_56(x))
Eq(F_56(x), F_25(x) + F_54(x))
Eq(F_57(x), 2*F_30(x) + F_58(x) + F_65(x))
Eq(F_58(x), F_16(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_16(x)*F_47(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), 2*F_30(x) + F_58(x) + F_64(x))
Eq(F_64(x), F_16(x)*F_54(x))
Eq(F_65(x), F_16(x)*F_66(x))
Eq(F_66(x), F_54(x))
Eq(F_67(x), F_16(x)*F_68(x))
Eq(F_68(x), F_24(x) + F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_40(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_16(x)*F_73(x))
Eq(F_73(x), F_69(x))
Pack JSON:
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Specification JSON:
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