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A301620 a(n) is the total number of top arches with exactly one covering arch for semi-meanders with n top arches.

%I A301620 #58 Apr 08 2021 03:41:15
%S A301620 0,0,2,4,18,42,156,398,1398,3778,12982,36522,124290,360182,1220440,
%T A301620 3618090,12237698,36938158,124880222,382471606,1293363816,4009185912,
%U A301620 13565790984,42478788432,143851766298,454339269482,1539997455570,4900091676662,16624834778474,53240459608298
%N A301620 a(n) is the total number of top arches with exactly one covering arch for semi-meanders with n top arches.
%C A301620 For n>2, a(n-2) is the number of ways to fold a strip of n stamps with leaf 1 on top and the n leaf not adjacent to the n-1 leaf. Example n = 6, a(6-2) = 4: 125436, 126345, 154362, 163452. - _Roger Ford_, Mar 29 2019
%C A301620 For n>2, a(n-2) is the number of ways to fold a strip of n stamps with leaf 1 on top and leaf 2 not in the second position and not in the n-th position. Example, for n = 6, a(6-2) = 4: 143265, 156234, 165234, 143256. - _Roger Ford_, Mar 12 2021
%H A301620 Jean-François Alcover, <a href="/A301620/b301620.txt">Table of n, a(n) for n = 1..43</a>
%F A301620 a(n) = A000682(n+2) - 2*A000682(n+1).
%F A301620 a(n) = Sum_{k=3..floor((n+3)/2)} (A259689(n+1,k)*(k-2)). - _Roger Ford_, Dec 10 2018
%F A301620 a(n) = 2*A259702(n+2). - _Roger Ford_, Dec 24 2018
%e A301620 For n = 4, a(4) = 4.  + + are underneath the starting and ending of each arch with exactly one covering arch.
%e A301620           /\                  /\
%e A301620          //\\         /\     //\\       /\
%e A301620       /\///\\\,  /\/\//\\,  ///\\\/\,  //\\/\/\ .
%e A301620          +  +         ++     +  +       ++
%t A301620 A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
%t A301620 a[n_] := A000682[[n + 2]] - 2*A000682[[n + 1]];
%t A301620 Array[a, 30] (* _Jean-François Alcover_, Sep 02 2019 *)
%Y A301620 Cf. A000682, A259689.
%K A301620 nonn
%O A301620 1,3
%A A301620 _Roger Ford_, Mar 24 2018