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%I #23 Oct 08 2017 09:56:55
%S 0,0,4,6,12,30,70,168,412,1014,2514,6270,15702,39468,99516,251586,
%T 637500,1618638,4117102,10488684,26758762,68354250,174810354,
%U 447533586,1146836662,2941443180,7550434480,19395863358,49859516292,128252962434,330101861850
%N Number of length-2n central circular binary strings without zigzags (see reference for precise definition).
%C See page 6 in the reference.
%C A zigzag is a substring which is either 010 or 101. The central binary strings are those that contain an equal number of 0's and 1's.
%H Andrew Howroyd, <a href="/A263656/b263656.txt">Table of n, a(n) for n = 0..100</a>
%H E. Munarini and N. Z. Salvi, <a href="http://www.emis.de/journals/INTEGERS/papers/d19/d19.Abstract.html">Circular Binary Strings without Zigzags</a>, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.
%F a(n) = (1/n)*(3*(n-1)*a(n-1) - 4*(n-4)*a(n-2) + (7*n-27)*a(n-3) - 6*a(n-4) + (7*n-37)*a(n-5) - 3*(n-6)*a(n-6)) for n >= 6. - _Andrew Howroyd_, Feb 26 2017
%e For n=3 the 6 strings are 000111, 001110, 011100, 111000, 110001, 100011.
%t a[n_ /; n < 6] := {0, 0, 4, 6, 12, 30}[[n + 1]]; a[n_] := a[n] = (-(3*(n - 6)*a[n - 6]) + (7*n - 37)*a[n - 5] - 6*a[n - 4] + (7*n - 27)*a[n - 3] - 4*(n - 4)*a[n - 2] + 3*(n - 1)*a[n - 1])/n;
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 08 2017, after _Andrew Howroyd_ *)
%Y Main diagonal of A263655.
%Y Cf. A007039, A263657, A263658, A263659.
%K nonn
%O 0,3
%A _Felix Fröhlich_, Oct 23 2015
%E corrected a(1) and a(17)-a(30) from _Andrew Howroyd_, Feb 26 2017