OFFSET
0,2
COMMENTS
The Ramanujan sequence number 11 for the argument 2Pi/9 defined by the relation a(n)*9^(1/3) = (3^(n-1))*(((-1)^(n-1))*(c(1) - 1/3)^(n + 1/3) + (((-1)^(n-1))*(c(2) - 1/3)^(n + 1/3) + (1/3 - c(4))^(n + 1/3)), where c(j) := 2*cos(2*Pi*j/9). The sequences A217052 and A217053 are conjugate with the sequence a(n). For more information on these connections - see Comments in A217053.
The 3-valuation of the sequence a(n) is equal to (0,2,1).
REFERENCES
R. Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012, in review.
LINKS
FORMULA
G.f.: x*(2+x)/(1-3*x-24*x^2-x^3).
EXAMPLE
We have a(4)-5*a(3)=32, 8*a(4)-a(5)=222, a(9)-a(6)=5226000. Furthermore from a(0)=0 we get (c(1) - 1/3)^( 1/3) + (c(2) - 1/3)^(1/3) = (1/3 - c(4))^(1/3), while from a(3)=69 we obtain 23*9^(-1/6) = (c(1) - 1/3)^(10/3) + (c(2) - 1/3)^(10/3) + (1/3 - c(4))^(10/3).
MATHEMATICA
LinearRecurrence[{3, 24, 1}, {0, 2, 7}, 30]
PROG
(PARI) Vec((2+x)/(1-3*x-24*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Sep 26 2012
STATUS
approved