OFFSET
0,1
COMMENTS
The Berndt-type sequence number 1 for the argument 2*Pi/9 (see also A215007, A215008) is connected with the following trigonometric identities: f(n;x)=g(n;x)=const for n=1,2 (and are equal to 6 and 18 respectively), f(n;x)+g(n;x)=const for n=3,4,5 (and are equal to 120, 420 and 1512 respectively). Moreover each of the functions f(3;x), g(3;x) and f(6;x)+g(6;x) is not the constant function. Here f(n;x) := (2*cos(x))^(2n) + (2*cos(x-Pi/3))^(2n) + (2*cos(x+Pi/3))^(2n), and g(n;x) := (2*sin(x))^(2n) + (2*cos(x-Pi/6))^(2n) + (2*cos(x+Pi/6))^(2n), for every n=1,2,..., and x in R (see Witula-Slota paper for details).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
R. Witula and D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl., 324 (2006), 321-343.
Index entries for linear recurrences with constant coefficients, signature (6,-9,1).
FORMULA
a(n) = c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n), where c(j) = 2*cos(Pi*j/9).
G.f.: 3*(1 - x)*(1 - 3*x)/(1 - 6*x + 9*x^2 - x^3).
a(n) = 3*A094831(n). - Andrew Howroyd, Apr 28 2020
EXAMPLE
From the identity c(j)^2 = 2 + c(2*j) we deduce that a(1)=6 is equivalent with c(2) + c(4) + c(8) = 0, where c(j) := 2*cos(Pi*j/9).
MATHEMATICA
LinearRecurrence[{6, -9, 1}, {3, 6, 18}, 50]
PROG
(PARI) Vec((3-12*x+9*x^2)/(1-6*x+9*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Aug 11 2012
EXTENSIONS
Terms a(22) and beyond from Andrew Howroyd, Apr 28 2020
STATUS
approved