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 A166289 #9 Jul 22 2022 12:54:56 %S A166289 1,1,1,2,2,4,6,9,17,26,46,81,135,246,428,757,1373,2431,4411,7990, %T A166289 14434,26423,48137,88144,162086,297662,549342,1014677,1876551,3480596, %U A166289 6458974,12008923,22361683,41675773,77797373,145368548,271917704 %N A166289 Number of Dyck paths with no UUU's and no DDD's, of semilength n and having no UDUD's (U=(1,1), D=(1,-1)). %C A166289 a(n) = A166288(n,0). %H A166289 Andrei Asinowski, Cyril Banderier, Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019). %F A166289 G.f.: G(z) satisfies z^3*G^2 - (1-z)(1+z)^2*G + (1+z)^2*G = 0. %F A166289 D-finite with recurrence +(n+3)*a(n) +(n+1)*a(n-1) -2*n*a(n-2) +2*(-3*n+5)*a(n-3) +(-3*n+11)*a(n-4) +(n-5)*a(n-5)=0. - _R. J. Mathar_, Jul 22 2022 %e A166289 a(5)=4 because we have UDUUDDUUDD, UUDDUDUUDD, UUDDUUDDUD, and UUDUUDDUDD. %p A166289 F := RootOf(z^3*G^2-(1-z)*(1+z)^2*G+(1+z)^2, G): Fser := series(F, z = 0, 40): seq(coeff(Fser, z, n), n = 0 .. 36); %Y A166289 Cf. A166288. %K A166289 nonn %O A166289 0,4 %A A166289 _Emeric Deutsch_, Oct 12 2009