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 A152171 #12 Jun 02 2025 01:13:08
%S A152171 1,1,1,2,5,12,29,73,190,505,1363,3727,10306,28771,80975,229512,654545,
%T A152171 1876899,5408142,15650939,45470545,132573406,387775229,1137575084,
%U A152171 3346189045,9867291817,29163523978,86377998093,256343194011
%N A152171 G.f. := (1-sqrt(1-4*x+4*x^2-4*x^3))/(2(1-x+x^2)x).
%C A152171 a(n) is the number of Dyck paths of length n without the height of peaks 2 (mod 3) and the height of valleys 1 (mod 3)
%H A152171 Michael De Vlieger, <a href="/A152171/b152171.txt">Table of n, a(n) for n = 0..2028</a>
%H A152171 Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/symasympyramid.pdf">Counting symmetric and asymmetric peaks in motzkin paths with air pockets</a>, Univ. Bourgogne (France, 2023).
%F A152171 D-finite with recurrence: (n+1)*a(n) +(-5*n+1)*a(n-1) +9*(n-1)*a(n-2) +12*(-n+2)*a(n-3) +2*(4*n-11)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2020
%t A152171 CoefficientList[Series[(1-Sqrt[1-4x+4x^2-4x^3])/(2(1-x+x^2)x),{x,0,40}],x] (* _Harvey P. Dale_, Feb 07 2022 *)
%K A152171 nonn
%O A152171 0,4
%A A152171 Jun Ma (majun(AT)math.sinica.edu.tw), Nov 27 2008
%E A152171 Offset corrected. - _R. J. Mathar_, Jan 25 2020