A215666 a(n) = 3*a(n-2) - a(n-3), with a(0)=0, a(1)=-3, and a(2)=6.

0, -3, 6, -9, 21, -33, 72, -120, 249, -432, 867, -1545, 3033, -5502, 10644, -19539, 37434, -69261, 131841, -245217, 464784, -867492, 1639569, -3067260, 5786199, -10841349, 20425857, -38310246, 72118920, -135356595, 254667006, -478188705, 899357613

Offset

0,2

Comments

The Berndt-type sequence number 7 for the argument 2Pi/9 defined by the first relation from the section "Formula" below. Two sequences connected with a(n) (possessing the respective numbers 5 and 6) are discussed in A215664 and A215665 - for more details see comments to A215664 and Witula's reference. We have a(n) = A215664(n+2) - 2*A215664(n) and a(n+1) = A215664(n+1) - A215664(n).

From initial values and the title recurrence formula we deduce that a(n)/3 and a(3*n)/9 are all integers.

If we set X(n) = 3*X(n-2) - X(n-3), n in Z, with a(n) = X(n), for every n=0,1,..., then X(-n) = -abs(A215917(n)) = (-1)^n*A215917(n), for every n=0,1,...

References

R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math., (in press, 2012).

D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the nine order, (submitted, 2012).

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (0, 3, -1).

Formula

a(n) = = c(4)*c(2)^n + c(1)*c(4)^n + c(2)*c(1)^n, where c(j):=2*cos(2*Pi*j/9).

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

Example

We have 8*a(3)+a(6)=5*a(6)+3*a(7)=0, a(5) + a(12) = 3000, and (a(30)-1000*a(10)-a(2))/10^5 is an integer. Further we obtain  c(4)*cos(4*Pi/7)^7 + c(1)*cos(8*Pi/7)^7 + c(2)*c(2*Pi/7)^7 = -15/16.

Mathematica

LinearRecurrence[{0, 3, -1}, {0, 3, -6}, 50].

Prog

(PARI) concat(0, Vec(-3*(1-2*x)/(1-3*x^2+x^3)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

Crossrefs

Cf. A215455, A215634, A215635, A215636, A215664, A214699, A215007, A214683.

Sequence in context: A015938 A116614 A089001 * A050889 A327140 A329863

Adjacent sequences: A215663 A215664 A215665 * A215667 A215668 A215669

Keyword

sign,easy

Author

Roman Witula, Aug 20 2012

Status

approved