A094255 Let M = the 3 X 3 matrix [ 0 1 0 / 0 0 1 / -1, 3*sqrt(3), 3]. M^n * [1 1 1] = [ p q r]; then a(n-1), a(n), a(n+1) = floor p, q, r, respectively.

1, 4, 9, 28, 75, 211, 577, 1591, 4367, 12004, 32975, 90607, 248931, 683946, 1879112, 5162835, 14184754, 38972316, 107075529, 294187633, 808273837, 2220714167, 6101361970, 16763354312, 46056937364, 126540395519

Offset

1,2

Comments

The matrix M is derived from a polynomial shown on p. 208 of "Advanced Trigonometry": sqrt(3)*x^3 - 3x^2 - 3*sqrt(3)*x + 1, which has roots tan 10 deg., tan 70 deg. and tan 130 deg. a(n)/a(n-1) tends to tan 70 deg. = 2.747477419...; e.g. a(11)/a(10) = 32975 / 12004 = 2.7470009...

References

C. V. Durell, A. Robson, "Advanced Trigonometry", Dover 2003, p. 208.

Links

Table of n, a(n) for n=1..26.

Example

a(4), a(5), a(6) = 28, 75, 211 = floor: p, q, r; where M^5 * [1 1 1] = [p q r].

Crossrefs

Sequence in context: A120333 A000368 A232765 * A192234 A069563 A352878

Adjacent sequences: A094252 A094253 A094254 * A094256 A094257 A094258

Keyword

nonn

Author

Gary W. Adamson, Apr 25 2004

Status

approved