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.
Chenyang (Amy) Hu, David A. Meyer, and Eleanor J. Q. Meyer, Reconstructing Minkowski geometry from causal separations, J. Math. Phys. 66 (2025) 122502.
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 - 4*x^2 + 5*x - 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
E.g.f.: exp(x)*(4*exp(x) - 3*x - 4). - Elmo R. Oliveira, Apr 01 2025
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,easy
AUTHOR
Gary W. Adamson, May 31 2004
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, Jun 05 2004
More terms from Elmo R. Oliveira, Apr 01 2025
STATUS
approved