A094253 Let M be the 3 X 3 Matrix [ -4 4 8 / 1 0 0 / 0 1 0], a(n) = absolute value of the center term of M^n * [1 1 1].

1, 8, 20, 120, 496, 2304, 10240, 46208, 207360, 932352, 4189184, 18827264, 84606976, 380223488, 1708703744, 7678853120, 34508439552, 155079540736, 696921096192, 3131935031296, 14074788184064, 63251524091904

Offset

1,2

Comments

The matrix is derived from the polynomial 8x^3 + 4x^2 - 4x - 1 shown on page 204 of "Advanced Trigonometry"; cos(2*Pi/7), cos(4*Pi/7), and cos(6*Pi/7) are roots of this polynomial.

References

C. V. Durell & A. Robson, "Advanced Trigonometry", Dover 2003, p. 204.

Links

Table of n, a(n) for n=1..22.

Index entries for linear recurrences with constant coefficients, signature (4, 4, -8).

Formula

a(n)/a(n-1) tends to 1/cos(3*Pi/7) = 4.4939592074...

From Colin Barker, Feb 01 2012: (Start)

a(n) = 4*a(n-1) + 4*a(n-2) - 8*a(n-3), n > 4.

G.f.: x*(1 + 4*x - 16*x^2 + 16*x^3)/(1 - 4*x - 4*x^2 + 8*x^3).

Example

a(3) = 20 since M^3 * [1 1 1] = [120 -20 8]. Take the absolute value of the center term.

Mathematica

Table[ Abs[ MatrixPower[{{-4, 4, 8}, {1, 0, 0}, {0, 1, 0}}, n].{1, 1, 1}][[2]], {n, 23}] (* Robert G. Wilson v, Apr 28 2004 *)

Crossrefs

Sequence in context: A074472 A175429 A297639 * A208084 A214905 A230987

Adjacent sequences: A094250 A094251 A094252 * A094254 A094255 A094256

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 25 2004

Extensions

More terms from Robert G. Wilson v, Apr 28 2004

Status

approved