This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A078679 #19 Jan 12 2025 12:30:59
%S A078679 1,1,3,7,17,43,111,291,771,2059,5533,14943,40523,110271,300949,823417,
%T A078679 2257877,6203239,17071779,47054475,129872499,358896927,992907525,
%U A078679 2749737663,7622185263,21146597511,58714466733,163142652877,453612137587,1262048222181,3513361583965
%N A078679 Number of Grand Motzkin paths of length n with no zigzags, that is with no factors UDU and DUD.
%C A078679 Also number of words on the alphabet {0,1,h} with length n, with an equal number of 1's and 0's and avoiding zigzags that is avoiding the subwords 101 and 010.
%H A078679 Emanuele Munarini and N. Z. Salvi, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s49zagaglia.html">Binary strings without zigzags</a>, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
%F A078679 G.f.: sqrt( ( 1 - x + x^2 ) / ( 1 - 3*x + x^3 + x^4 ) ).
%F A078679 Recurrence: 0 = (n+6)*a(n+6) - (4*n+21)*a(n+5) + (4*n+15)*a(n+4) - (2*n+3)*a(n+3) + a(n+2) - a(n+1) + (n+1)*a(n).
%e A078679 For n = 3 we have the words hhh, 01h, 0h1, h01, 10h, 1h0, h10.
%t A078679 Table[SeriesCoefficient[Series[Sqrt[ ( 1 - x + x^2 ) / ( 1 - 3 x + x^3 + x^4 )], {x, 0, n}], n], {n, 0, 40}]
%o A078679 (Maxima) a(n):=coeff(taylor(sqrt((1-x+x^2)/(1-3*x+x^3+x^4)),x,0,n),x,n);
%o A078679 makelist(a(n),n,0,12); /* _Emanuele Munarini_, Jul 07 2011 */
%Y A078679 Cf. A078678.
%K A078679 easy,nonn
%O A078679 0,3
%A A078679 _Emanuele Munarini_, Dec 17 2002