This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A095264 #43 Apr 02 2025 18:32:11
%S A095264 1,6,19,48,109,234,487,996,2017,4062,8155,16344,32725,65490,131023,
%T A095264 262092,524233,1048518,2097091,4194240,8388541,16777146,33554359,
%U A095264 67108788,134217649,268435374,536870827,1073741736,2147483557,4294967202,8589934495,17179869084,34359738265
%N A095264 a(n) = 2^(n+2) - 3*n - 4.
%C A095264 A sequence derived from a 3rd-order matrix generator.
%C A095264 The number of positive 3-strand braids of degree at most n. - _R. J. Mathar_, May 04 2006
%C A095264 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
%H A095264 P. Dehornoy, <a href="https://arxiv.org/abs/math/0511114">Combinatorics of normal sequences of braids</a>, arXiv:math/0511114 [math.CO], 2005.
%H A095264 Shishuo Fu, Zhicong Lin, and Yaling Wang, <a href="https://arxiv.org/abs/2009.04269">Refined Wilf-equivalences by Comtet statistics</a>, arXiv:2009.04269 [math.CO], 2020.
%H A095264 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A095264 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.
%F A095264 a(n+3) = 4*a(n+2) - 5*a(n+1) + 2*a(n).
%F A095264 a(n) = Sum_{i=2..n+1} A036563(i) [A036563 is 2^i-3]. - _Gerald McGarvey_, Jun 28 2004
%F A095264 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
%F A095264 a(n) = 2*a(n-1) + (3n-2). - _Gary W. Adamson_, Sep 30 2007
%F A095264 G.f.: -x*(1+2*x)/((2*x-1)*(x-1)^2). - _R. J. Mathar_, Nov 18 2007
%F A095264 E.g.f.: exp(x)*(4*exp(x) - 3*x - 4). - _Elmo R. Oliveira_, Apr 01 2025
%e A095264 a(5) = 109 = 2^7 - 3*5 - 4.
%e A095264 a(5) = 109 since M^5 * [1 0 0] = [1 5 109].
%e A095264 a(7) = 487 = 4*234 - 5*109 + 2*48.
%t A095264 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 *)
%t A095264 Table[2^(n+2)-3n-4,{n,40}] (* or *) LinearRecurrence[{4,-5,2},{1,6,19},40] (* _Harvey P. Dale_, Sep 24 2021 *)
%Y A095264 Cf. A036563, A125232, A126277.
%K A095264 nonn,easy
%O A095264 1,2
%A A095264 _Gary W. Adamson_, May 31 2004
%E A095264 Edited, corrected and extended by _Robert G. Wilson v_, Jun 05 2004
%E A095264 More terms from _Elmo R. Oliveira_, Apr 01 2025