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 A375995 #38 Mar 04 2025 08:35:48
%S A375995 0,0,0,0,1,7,30,109,365,1164,3593,10835,32106,93845,271321,777432,
%T A375995 2211025,6248479,17562870,49132669,136884293,379975140,1051356761,
%U A375995 2900587115,7981564866,21911096357,60021530545,164095925424,447823729825,1220105286199,3319124711118
%N A375995 Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.
%C A375995 A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.
%H A375995 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 A375995 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 A375995 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. 15, 19.
%H A375995 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 A375995 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6, -11, 6, -1).
%F A375995 a(n) = (2(n-3)*L(2 n-5)-3F(2n-6))/5 for n>=3 and a(n) = 0 for n<=2, F(.) is a Fibonacci number, L(.) is a Lucas number.
%F A375995 G.f.: x^4*(-x^2+x+1)/(x^2-3x+1)^2.
%t A375995 Table[If[n<=2,0,(2(n-3)LucasL[2n-5]-3Fibonacci[2n-6])/5], {n,0,30}]
%Y A375995 Cf. A000032, A000045, A377679, A377670.
%K A375995 nonn,easy
%O A375995 0,6
%A A375995 _Rigoberto Florez_, Nov 03 2024