1,2

A sequence derived from a 3rd order matrix generator.

The number of positive 3-strand braids of degree at most n. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 04 2006

P. Dehornoy, Combinatorics of normal sequences of braids, /math.CO/0511114.

Let M = [1 0 0 / 1 1 0 / 1 3 2], then M^n * [1 0 0] = [1 n a(n)]. The characteristic polynomial of M is x^3 - 4x^2 + 5x - 2.

a(n+3) = 4*a(n+2) - 5*a(n+1) + 2*a(n).

a(n) = Sum i=2..n+1 A036563(i) [A036563 is 2^i-3] - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 28 2004

Row sums of A125232; 5-th diagonal from the right of A126277; binomial transform of [1, 5, 8, 8, 8,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2006

a(n) = 2*a(n-1) + (3n-2). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 30 2007

G.f.: -x*(1+2*x)/(-1+x)^2/(-1+2*x) = -4/(-1+2*x)-3/(-1+x)^2+1/(-1+x) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 18 2007

a(5) = 109 = 2^7 - 3*5 - 4.

a(5) = 109 since M^5 * [1 0 0] = [1 5 109].

a(7) = 487 = 4*234 - 5*109 + 2*48.

a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {1, 3, 2}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 30}] (from Robert G. Wilson v Jun 05 2004)

Cf. A036563.

Cf. A126277, A125232.

Sequence in context: A027963 A034199 A073362 this_sequence A027044 A057571 A125069

Adjacent sequences: A095261 A095262 A095263 this_sequence A095265 A095266 A095267

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), May 31 2004

Edited, corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 05 2004