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A201631 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 2.

%I A201631 #31 Mar 28 2023 12:50:13
%S A201631 1,3,6,13,30,70,167,405,992,2450,6090,15214,38165,96069,242530,613811,
%T A201631 1556856,3956316,10070871,25674210,65541142,167517654,428635032,
%U A201631 1097874434,2814611701,7221917871,18544968768,47655572191,122544150258,315313433594,811792614547
%N A201631 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 2.
%C A201631 Empirically the partial sums of A051291. - _Sean A. Irvine_, Jul 13 2022
%C A201631 The above conjecture was proved by Baril et al., which also give a formal definition of the Fibonacci meanders and describe a bijection with a certain class of peakless grand Motzkin paths of length n. - _Peter Luschny_, Mar 16 2023
%H A201631 Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/2202.06893">Enumeration of Dyck paths with air pockets</a>, arXiv:2202.06893 [cs.DM], 2022-2023.
%H A201631 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FibonacciMeanders">Fibonacci meanders</a>.
%F A201631 a(n) = Sum_{k=0..2n-1} A202411(k).
%F A201631 a(n) = [x^n] (x^2 - x + 1 - R)/((x - 1)*(x^2 - x - 1 + R) * R), where R = (((x - 3)*x + 1)*(x^2 + x + 1))^(1/2). (This is Theorem 21 in Baril et al.) - _Peter Luschny_, Mar 16 2023
%e A201631 a(3) = 6 = card({100001, 100100, 110000, 111001, 111100, 111111}).
%p A201631 A201631 := n -> add(A202411(k),k=0..2*n-1): seq(A201631(i),i=1..9);
%p A201631 # Alternative, using the g.f. of Baril et al.:
%p A201631 S := (x^2 - x + 1 - R)/((x - 1)*(x^2 - x - 1 + R)*R):
%p A201631 R := (((x - 3)*x + 1)*(x^2 + x + 1))^(1/2): ser := series(S, x, 33):
%p A201631 seq(coeff(ser, x, n), n = 1..31); # _Peter Luschny_, Mar 16 2023
%p A201631 # Using a recurrence:
%p A201631 a := proc(n) option remember; if n < 5 then return [0, 1, 3, 6, 13][n + 1] fi;
%p A201631 (n*(2*n - 1)*(2*n - 3)*(n - 5)*a(n - 5) - (n - 4)*(2*n - 1)^2*(3*n - 5)*a(n - 4) + (2*n - 5)*(n - 3)*(2*n^2 - 3*n + 2)*a(n - 3) - (2*n - 3)*(n - 2)*(2*n^2 - 3*n + 5)*a(n - 2) + (3*n - 4)*(2*n - 1)*(2*n - 5)*(n - 1)*a(n - 1))/(n*(2*n - 3)*(2*n - 5)*(n - 1)) end: seq(a(n), n = 1..31); # _Peter Luschny_, Mar 16 2023
%t A201631 a[n_] := Sum[A202411[k], {k, 0, 2 n - 1}];
%t A201631 Array[a, 31] (* _Jean-François Alcover_, Jun 29 2019 *)
%Y A201631 Cf. A110236, A110198, A202411, A203611, A051291, A358734.
%Y A201631 Cf. A361574 (case m=3).
%K A201631 nonn
%O A201631 1,2
%A A201631 _Peter Luschny_, Jan 15 2012