A217069 a(n) = 3*a(n-1) + 24*a(n-2) + a(n-3), with a(0)=0, a(1)=2, and a(2)=7.

0, 2, 7, 69, 377, 2794, 17499, 119930, 782560, 5243499, 34631867, 230522137, 1527974718, 10151087309, 67355177296, 447219602022, 2968334148479, 19705628071261, 130804123379301, 868315777996646, 5763951923164423, 38262238564792074, 253989877628319020

Offset: 0

Author

Roman Witula, Sep 26 2012

Keywords

• nonn
• easy

Comments

The Ramanujan sequence number 11 for the argument 2*Pi/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).

Examples

			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).
		

References

• Roman 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).

Crossrefs

Cf. A217052, A217053, A214699, A214779, A214778, A214951, A214954, A215560, A215569, A215572.

Programs

Mathematica

LinearRecurrence[{3,24,1}, {0,2,7}, 30]

PARI

Vec((2+x)/(1-3*x-24*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

G.f.: x*(2+x)/(1-3*x-24*x^2-x^3).