A094257 Let M be the 3 X 3 matrix [0 1 0 / 0 0 1 / 7 -35 21]. Take M^n * [1 1 1] = [p q r]; then a(n-1), a(n), a(n+1) = -p, -q, -r respectively.

1, 7, 175, 3423, 65807, 1263367, 24251423, 465522687, 8936020191, 171532889927, 3292688640591, 63205362446303, 1213269239181167, 23289515157668039, 447057832476812095, 8581574336168940799, 164729063528963009727

Offset

1,2

Comments

a(n)/a(n-1) tends to tan^2(3*Pi/7) = 19.195669358...

The matrix M has an eigenvalue of tan^2(3*Pi/7) which is one root of x^3 - 21*x^2 + 35*x - 7 (the two other roots being tan^2(Pi/7) and tan^2(2*Pi/7)).

References

C. V. Durell and A. Robson, "Advanced Trigonometry", Dover 2003, p. 205.

Links

Table of n, a(n) for n=1..17.

Index entries for linear recurrences with constant coefficients, signature (21,-35,7).

Formula

G.f.: x*(14*x^3 - 63*x^2 + 14*x - 1)/(7*x^3 - 35*x^2 + 21*x - 1). [Colin Barker, Nov 08 2012]

Example

a(4), a(5), a(6) = 3423, 65807, 1263367 since M^5 * [1 1 1] = [ -3423 -65807 -1263367].

Mathematica

Table[ Abs[ MatrixPower[{{0, 1, 0}, {0, 0, 1}, {7, -35, 21}}, n].{1, 1, 1}][[2]], {n, 17}] (* Robert G. Wilson v, Apr 28 2004 *)

Crossrefs

Sequence in context: A075599 A012500 A220326 * A195887 A162082 A195517

Adjacent sequences: A094254 A094255 A094256 * A094258 A094259 A094260

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 25 2004

Extensions

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

Status

approved