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 A377679 #28 Jul 18 2025 01:53:08
%S A377679 0,0,0,1,6,26,97,333,1085,3411,10448,31376,92773,270907,783003,
%T A377679 2243815,6383550,18048494,50755897,142067625,396014681,1099863867,
%U A377679 3044737100,8404071596,23135752141,63538808311,174120317367,476207551183
%N A377679 Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.
%C A377679 A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
%H A377679 E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170 (1997), 211-217.
%H A377679 Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
%H A377679 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 6, 19.
%H A377679 Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
%H A377679 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (8,-23,28,-13,2).
%F A377679 a(n) = n*F(2*n-3) - L(2*n-2) + 2^(n-2) for n>=2, where F(n) = A000045(n) and L(n) = A000032(n).
%F A377679 G.f.: x^3*(1 - 2*x + x^2 - x^3)/((1 - 2*x)*(1 - 3*x + x^2)^2).
%t A377679 Table[If[n<2,0,n Fibonacci[2 n-3]-LucasL[2 n-2]+2^(n-2)],{n,0,30}]
%Y A377679 Cf. A000032, A000045, A377670, A375995.
%K A377679 nonn,easy
%O A377679 0,5
%A A377679 _Rigoberto Florez_, Nov 03 2024