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<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6, -9, 1).

R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl., 324 (2006), 321-343.

R. Witula and D. Slota, <a href="http://dx.doi.org/10.1016/j.jmaa.2005.12.020">On modified Chebyshev polynomials</a>, J. Math. Anal. Appl., 324 (2006), 321-343.

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The Berndt-type sequence number 1 for the argument 2Pi2*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).

a(n) = c(1)^(2n2*n) + c(2)^(2n2*n) + c(4)^(2n2*n), where c(j) = 2*cos(Pi*j/9).

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 like above we have c(j) := 2*cos(Pi*j/9).

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3, 6, 18, 57, 186, 621, 2109, 7251, 25146, 87726, 307293, 1079370, 3798309, 13382817, 47191491, 166501902, 587670810, 2074699233, 7325660010, 25869337773, 91359785781, 322660334739, 1139593274178, 4024976418198, 14216179376325, 50211881768346, 177350652641349

The Berndt-type sequence number 1 for the argument 2Pi/9 (see also A215007, A215008) is connect 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).

Andrew Howroyd, <a href="/A215455/b215455.txt">Table of n, a(n) for n = 0..500</a>

G.f.:( 3-12*(1 - x+9)*(1 - 3*x^2)/(1 - 6*x + 9*x^2 - x^3).

a(n) = 3*A094831(n). - Andrew Howroyd, Apr 28 2020

Cf. A094831, A215007, A215008.

Terms a(22) and beyond from Andrew Howroyd, Apr 28 2020

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