%I #25 Sep 29 2021 10:55:19
%S 0,2,2,12,20,88,180,716,1648,6244,15512,57188,149892,543324,1481636,
%T 5310764,4930380,53102508,152935148,540918660,1588618212,5595773136,
%U 16701973552
%N a(n) is the number of arches with a length of one and exactly one covering arch for semi-meanders with n top arches.
%C The ratio of the number of semi-meanders in generation n+1 to the number of semi-meanders in generation n is equal to the ratio of the differences of these semi-meander subset arch sequences in successive generations.
%C A000682(n+1)/A000682(n) = (A260785(n+3) - a(n+1))/(A260785(n+2) - a(n)).
%D See A000682.
%H P. Di Francesco, O. Golinelli and E. Guitter, <a href="https://arxiv.org/abs/cond-mat/9910453">Meanders: exact asymptotics</a>, arXiv:cond-mat/9910453 [cond-mat.stat-mech], 1999-2000; Nuclear Physics B, volume 570, issue 3, 27 March 2000, 699-712.
%F For n >= 2, a(n) = A260785(n+2) - 2*A000682(n).
%e n = 5, || indicates an arch of length one with exactly one covering arch.
%e /\ /\ /\
%e /\ //\\ //\\ /\ /\ /\ /\ /\ / \
%e //\\ ///\\\ ///\\\ //\\ /\ //\\ //\\ //\\ //\\ /\ /\ /\ //\/\\
%e || || || || || || ||||
%e /\ /\
%e / \ / \ /\
%e //\ \ / /\\ / \
%e ///\\/\\ /\ /\ //\//\\\ //\/\\ /\ /\
%e || || |||| a(5) = 12.
%Y Cf. A000682, A260785.
%K nonn,more
%O 2,2
%A _Roger Ford_, Aug 01 2021