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 A340569 #12 Feb 20 2021 04:02:39
%S A340569 0,0,0,1,4,10,24,53,116,246,520,1082,2248,4628,9520,19469,39796,81022,
%T A340569 164904,334670,679064,1374924,2783440,5625666,11368904,22945820,
%U A340569 46307664,93358228,188202256,379078952,763506784,1536708413,3092806516,6220970702,12512656744,25154958278
%N A340569 Total number of consecutive triples matching the pattern 123 in all faro permutations of length n.
%C A340569 Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one.
%H A340569 Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, <a href="https://arxiv.org/abs/2010.06270">Pattern statistics in faro words and permutations</a>, arXiv:2010.06270 [math.CO], 2020. See Table 1.
%F A340569 G.f.: x * (1+2*x) * (1-sqrt(1-4*x^2)) / ((1-2*x) * (1+sqrt(1-4*x^2))).
%e A340569 For n = 4, there are 6 faro permutations: 1234, 1243, 1324, 2134, 2143, 3142. They contain in total 4 consecutive patterns 123.
%t A340569 Table[SeriesCoefficient[x*(1+2*x)*(1-Sqrt[1-4*x^2])/((1-2*x) * (1+Sqrt[1-4*x^2])),{x,0,n}],{n,0,35}] (* _Stefano Spezia_, Jan 12 2021 *)
%Y A340569 A001405 counts faro permutations of length n.
%Y A340569 Cf. A107373, A340567, A340568.
%K A340569 nonn
%O A340569 0,5
%A A340569 _Sergey Kirgizov_, Jan 12 2021