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 A364685 #26 Aug 05 2023 21:27:47
%S A364685 4,10,18,30,48,76,120,190,302,482,772,1240,1996,3218,5194,8390,13560,
%T A364685 21924,35456,57350,92774,150090,242828,392880,635668,1028506,1664130,
%U A364685 2692590,4356672,7049212,11405832,18454990,29860766,48315698,78176404,126492040,204668380
%N A364685 The number of binary sequences of length n for which all patterns {0,1},{0,0},{1,0},{1,1} appear for the first time. In particular, three of the patterns will have appeared at least once before the (n-1)st digit in the sequence and the remaining pattern appears for the first and only time at positions {n-1,n} in the sequence.
%H A364685 Evan Fisher, Tianman Huang, and Xiaoshi Wang, <a href="https://doi.org/10.1216/rmj.2020.50.957">Cover Times for Markov Generated Sequences of Length Two</a>, Rocky Mountain J. Math. 50 (3) 957 - 974, June 2020.
%H A364685 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A364685 a(n) = 2*(n-6+F(n-1)), F(n) is the n-th Fibonacci number A000045(n).
%F A364685 G.f.: 2*x^5*(2*x^2+x-2)/((x^2+x-1)*(x-1)^2).
%e A364685 a(6)=10 is the number of cover time sequences of length 6 for binary patterns of length 2: {{0, 0, 0, 1, 1, 0}, {0, 0, 1, 0, 1, 1}, {0, 0, 1, 1, 1, 0}, {0, 1, 0, 0, 1, 1}, {0, 1, 1, 1, 0, 0}, {1, 0, 0, 0, 1, 1}, {1, 0, 1, 1, 0, 0}, {1, 1, 0, 0, 0, 1}, {1, 1, 0, 1, 0, 0}, {1, 1, 1, 0, 0, 1}}. (Notice that the final two digits in each of these sequences completes the appearance of all four patterns.)
%t A364685 b[n_]:= b[n] = Tuples[{0, 1}, n];
%t A364685 a1[n_]:=
%t A364685 Select[b[n],
%t A364685 MatchQ[#, {___, PatternSequence[0, 0], ___}] &&
%t A364685 MatchQ[#, {___, PatternSequence[0, 1], ___}] &&
%t A364685 MatchQ[#, {___, PatternSequence[1, 0], ___}] &&
%t A364685 MatchQ[#, {___, PatternSequence[1, 1], ___}] &];
%t A364685 Table[Length[
%t A364685 Select[a1[k], Length[SequencePosition[#, Take[#, -2]]] == 1 &]], {k,
%t A364685 5, 20}]
%Y A364685 Cf. A000045, A242206.
%K A364685 nonn,easy
%O A364685 5,1
%A A364685 _Evan Fisher_ and Ruiqi (Violet) Cai, Aug 02 2023