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 A332345 #7 Feb 11 2020 12:07:05
%S A332345 1,1,2,4,6,12,28,76,240,852,3392,14868,71392,371740,2089184,12589980,
%T A332345 81037792,554555300,4021728992,30800978980,248464000480,2105271605804,
%U A332345 18696216008416,173630895701996,1683187452989920,17000599430003444,178625854452674272
%N A332345 a(n) is the number of totally alternating permutations of 1,2,...,n.
%C A332345 A permutation w of 1,...,n is totally alternating if both w and w^{-1} are either alternating or reverse alternating.
%C A332345 Totally alternating permutations are those that avoid the consecutive patterns 123 and 321 and also the consecutive covincular patterns 123 and 321.
%C A332345 If a permutation is totally alternating, then so are its other 7 symmetries.
%H A332345 R. P. Stanley, <a href="http://arxiv.org/abs/math/0603520">Alternating permutations and symmetric functions</a>, arXiv:math/0603520 [math.CO], 2006.
%e A332345 The six totally alternating permutations of 1,...,4 are 1324, 2143, 2413, 3142, 3412 and 4231.
%Y A332345 For n > 1, a(n) = 2*(A007999(n) + A332344(n)).
%Y A332345 For odd n > 1, a(n) = 4*A007999(n).
%Y A332345 For even n > 1, a(n) = 4*A007999(n) - 2*A007999(n-2).
%K A332345 nonn
%O A332345 0,3
%A A332345 _David Bevan_, Feb 10 2020