OFFSET
0,3
COMMENTS
The Ramanujan type sequence number 10 for the argument 2Pi/9 defined by the relation a(n) = ((1/3 - c(1))^n + (1/3 - c(2))^n + (1/3 - c(4))^n)*3^(n-1), where c(j) := 2*cos(2*Pi*j/9). We note that c(4) = -cos(Pi/9). The conjugate with a(n) are sequences A217053 and A217069.
For more informations about connections a(n) with these two sequences - see comments in A217053.
The 3-valuation of the sequence a(n) is equal to (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
Index entries for linear recurrences with constant coefficients, signature (3, 24, 1).
FORMULA
G.f.: (1-2*x-8*x^2)/(1-3*x-24*x^2-x^3).
EXAMPLE
We have a(4)=37*a(2) and a(5) = 2^(12), which implies ((1/3 - c(1))^4 + (1/3 - c(2))^4 + (1/3 - c(4))^4 = (37/9)*((1/3 - c(1))^2 + (1/3 - c(2))^2 + (1/3 - c(4))^2) = (37/27)*19 = 703/27, (1/3 - c(1))^5 + (1/3 - c(2))^5 + (1/3 - c(4))^5 = (8/3)^4. Moreover we have a(10) = 676837*a(3)
MATHEMATICA
LinearRecurrence[{3, 24, 1}, {1, 1, 19}, 30]
PROG
(PARI) Vec((1-2*x-8*x^2)/(1-3*x-24*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Sep 25 2012
STATUS
approved