OFFSET
1,3
COMMENTS
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).
LINKS
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
FORMULA
G.f.: x*(1-x+x^2)^2/(1-x-x^3)^2.
EXAMPLE
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.
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.
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.
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Steven Finch, May 26 2020
STATUS
approved