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 A209232 #25 Jan 21 2023 02:18:20
%S A209232 0,1,4,11,25,52,103,199,380,724,1382,2649,5103,9881,19224,37559,73646,
%T A209232 144848,285623,564429,1117396,2215436,4398054,8740266,17385207,
%U A209232 34607218,68934319,137386725,273942683,546450648,1090419638
%N A209232 a(n) is 2^n times the expected value of the shortest run of 0's in a binary word of length n.
%C A209232 a(n) is also the sum of the number of binary words containing at least one 0 and having every consecutive run of 0's of length >= i for i >= 1. In other words, a(n) = A000225(n) + A077855(n) + A130578(n) + A209231(n) + ...
%D A209232 R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, Chapter 7.
%H A209232 G. C. Greubel, <a href="/A209232/b209232.txt">Table of n, a(n) for n = 0..1000</a>
%F A209232 O.g.f.: Sum_{k >= 1} (x^k/(1 - x) + 1) / ((1 - x^(k + 1)/(1 - x)^2)) * 1/(1 - x) - 1/(1 - x).
%e A209232 a(3) = 11. To the length 3 binary words {0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1} we have respectively shortest zero runs of length 3 + 2 + 1 + 1 + 2 + 1 + 1 + 0 = 11.
%t A209232 nn = 30; Apply[Plus, Table[a = x^n/(1 - x); CoefficientList[Series[(a + 1)/((1 - a x/(1 - x)))*1/(1 - x) - 1/(1 - x), {x, 0, nn}], x], {n, 1, nn}]]
%Y A209232 Cf. A119706.
%K A209232 nonn
%O A209232 0,3
%A A209232 _Geoffrey Critzer_, Jan 12 2013