OFFSET
1,2
COMMENTS
A sequence derived from a 3rd-order matrix generator.
The number of positive 3-strand braids of degree at most n. - R. J. Mathar, May 04 2006
Define a triangle T by T(n,n) = n*(n+1)/2, T(n,1) = n*(n-1) + 1, and T(r,c) = T(r-1,c-1) + T(r-1,c). Its rows are 1; 3,3; 7,6,6; 13,13,12,10; 21,26,25,22,15; etc. The sum of the terms in the n-th row is a(n). - J. M. Bergot, May 03 2013
LINKS
P. Dehornoy, Combinatorics of normal sequences of braids, arXiv:math/0511114 [math.CO], 2005.
Shishuo Fu, Zhicong Lin, and Yaling Wang, Refined Wilf-equivalences by Comtet statistics, arXiv:2009.04269 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
FORMULA
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).
Row sums of A125232; 5th diagonal from the right of A126277; binomial transform of [1, 5, 8, 8, 8, ...]. - Gary W. Adamson, Dec 23 2006
a(n) = 2*a(n-1) + (3n-2). - Gary W. Adamson, Sep 30 2007
G.f.: -x*(1+2*x)/((2*x-1)*(x-1)^2). - R. J. Mathar, Nov 18 2007
EXAMPLE
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.
MATHEMATICA
a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {1, 3, 2}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 30}] (* Robert G. Wilson v, Jun 05 2004 *)
Table[2^(n+2)-3n-4, {n, 40}] (* or *) LinearRecurrence[{4, -5, 2}, {1, 6, 19}, 40] (* Harvey P. Dale, Sep 24 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, May 31 2004
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, Jun 05 2004
STATUS
approved