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 A373567 #30 Apr 14 2025 23:12:05
%S A373567 1,4,6,14,31,70,157,353,793,1782,4004,8997,20216,45425,102069,229347,
%T A373567 515338,1157954,2601899,5846414,13136773,29518061,66326481,149034250,
%U A373567 334876920,752461609,1690765888,3799116465,8536537209,19181424995,43100270734,96845429254
%N A373567 Expansion of x + 1/(-x - 1/(-x - 1/(-x + 1))).
%C A373567 a(n) is the number of up-down words of length n over an alphabet of size 4. - _Sela Fried_, Apr 08 2025
%D A373567 L. Carlitz and R. Scoville, Up-down sequences, Duke Math. J. (39) (1972), 583-598.
%H A373567 Sela Fried, <a href="https://arxiv.org/abs/2503.02005">A formula for the number of up-down words</a>, arXiv:2503.02005 [math.CO], 2025.
%H A373567 Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, <a href="https://arxiv.org/abs/1505.04078">Pattern-avoiding alternating words</a>, arXiv:1505.04078 [math.CO], 2015.
%H A373567 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1).
%F A373567 a(n) = [x^n] (x^4 - x^3 - 3*x^2 + 2*x + 1) / (x^3 - x^2 - 2*x + 1).
%t A373567 CoefficientList[Series[x + 1/(-x - 1/(-x - 1/(-x + 1))), {x, 0, 31}], x] (* _Michael De Vlieger_, Jun 10 2024 *)
%Y A373567 Essentially the same as A006356.
%Y A373567 Cf. A050446.
%K A373567 nonn,easy
%O A373567 0,2
%A A373567 _Peter Luschny_, Jun 10 2024