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 A307909 #7 Jul 29 2019 13:17:30 %S A307909 1,1,2,3,5,8,15,27,52,99,195,382,757,1499,2986,5945,11865,23678,47309, %T A307909 94519,188942,377689,755191,1510000,3019625,6038493,12076244,24150989, %U A307909 48300491,96597996,193193033,386380121,772754322,1545496779,3090981745,6181939812 %N A307909 Number of binary strings of length n whose only palindromic prefixes are "1" and "11". %C A307909 Obviously, any 1-length prefix will be palindromic. Without the 2-length prefix condition, there is only one such string for every length: 10, 100, 1000, etc. (A000012). %F A307909 a(2) = 1, a(3) = 1, a(2*k) = 2*a(2*k-1)-a(k+1)+a(k), a(2*k+1) = 2*a(2*k)-a(k+1) %e A307909 a(5) = 3 because there are three such strings: 11000, 11001, and 11010. For example, 11100 is not such a string, because a prefix (111) is palindromic. %Y A307909 Cf. A000012, A006995. %K A307909 nonn,easy %O A307909 2,3 %A A307909 _Lewis Chen_, May 04 2019