

A215666


a(n) = 3*a(n2)  a(n3), with a(0)=0, a(1)=3, and a(2)=6.


9



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

The Berndttype 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(n2)  X(n3), 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 quasiFibonacci 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*(12*x)/(13*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*(12*x)/(13*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



