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