OFFSET
1,2
COMMENTS
A bitstring is multus if each of its 1's possess at least one neighboring 1.
The number of these bitstrings is A005251(n+2).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..985
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
FORMULA
G.f.: -x/((1-x)*(1-x+x^2)) + x*Sum_{k>=1} (1+x^2)/(1-2*x+x^2-x^3) - (1+x^2-x^(k-1)-x^k)/(1-2*x+x^2-x^3+x^(k+1)).
EXAMPLE
a(4) = 16 because the seven multus bitstrings of length 4 are 0000, 1100, 0110, 0011, 1110, 0111, 1111 and the longest 1-runs contribute 0+2+2+2+3+3+4 = 16.
MATHEMATICA
gf[n_] := x/((x - 1) (1 - x + x^2)) + Sum[((x - 1) x^k)/((x^3 - x^2 + 2 x - 1) (x^(k + 1) - x^3 + x^2 - 2 x + 1)), {k, 1, n}];
ser[n_] := Series[gf[n], {x, 0, n}];
Drop[CoefficientList[ser[36], x], 1] (* Peter Luschny, Mar 19 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Steven Finch, Mar 18 2020
STATUS
approved