A095264 a(n) = 2^(n+2) - 3*n - 4.

1, 6, 19, 48, 109, 234, 487, 996, 2017, 4062, 8155, 16344, 32725, 65490, 131023, 262092, 524233, 1048518, 2097091, 4194240, 8388541, 16777146, 33554359, 67108788, 134217649, 268435374, 536870827, 1073741736, 2147483557, 4294967202, 8589934495, 17179869084, 34359738265

Offset: 1

Author

Gary W. Adamson, May 31 2004

Keywords

• nonn
• easy

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

Examples

			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.
		

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).

Crossrefs

Cf. A036563, A125232, A126277.

Programs

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 *)

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).

a(n) = Sum_{i=2..n+1} A036563(i) [A036563 is 2^i-3]. - Gerald McGarvey, Jun 28 2004

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

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 05 2004

More terms from Elmo R. Oliveira, Apr 01 2025