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%I #7 May 26 2020 12:05:32
%S 1,0,2,4,5,10,18,28,46,76,121,192,305,480,751,1172,1822,2822,4359,
%T 6716,10322,15830,24230,37020,56467,85998,130787,198640,301325,456568,
%U 691050,1044904,1578457,2382334,3592594,5413392,8150894,12264012,18440269,27709196
%N a(n) is the total binary weight of all persolus bitstrings of length n.
%C A bitstring is persolus if all of its 1's are isolated and each of its 0's possess at least one neighboring 0. The number of persolus bitstrings of length n is A179070(n+1).
%H Steven Finch, <a href="https://arxiv.org/abs/2005.12185">Variance of longest run duration in a random bitstring</a>, arXiv:2005.12185 [math.CO], 2020.
%F G.f.: x*(1-x+x^2)^2/(1-x-x^3)^2.
%e The only three persolus bitstrings of length 3 are 000, 100 and 001. The bitsums of these are 0, 1 and 1. Adding these give a(3)=2.
%e The only four persolus bitstrings of length 4 are 0000, 1000, 0001, and 1001. The bitsums of these are 0, 1, 1, and 2. Adding these give a(4)=4.
%e The only five persolus bitstrings of length 5 are 00000, 10000, 00100, 00001, and 10001. The bitsums of these are 0, 1, 1, 1 and 2. Adding these give a(5)=5.
%e The only eight persolus bitstrings of length 6 are 000000, 100000, 001000, 000100, 000001, 100100, 100001, and 001001. The bitsums of these are 0, 1, 1, 1, 1, 2, 2 and 2. Adding these give a(6)=10.
%Y Cf. A001629, A179070.
%K nonn
%O 1,3
%A _Steven Finch_, May 26 2020