A333396 - OEIS

%I #29 Mar 24 2020 19:09:04

%S 1,2,5,11,23,45,87,165,309,573,1056,1934,3527,6408,11605,20960,37771,

%T 67928,121949,218595,391302,699610,1249475,2229329,3974083,7078658,

%U 12599318,22410548,39837420,70775727,125675525,223052519,395702395,701695820,1243827018,2204007329

%N Total length of all longest runs of 0's in multus bitstrings of length n.

%C A bitstring is multus if each of its 1's possess at least one neighboring 1.

%C The number of these bitstrings is A005251(n+2).

%H Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.

%F G.f.: x*Sum_{k>=1} (1+x^2)/(1-2*x+x^2-x^3)-(1+x^2-x^(k-1)+x^k-2*x^(k+1))/(1-2*x+x^2-x^3+x^(k+2)).

%e a(4) = 11 because the seven multus bitstrings of length 4 are 0000, 1100, 0110, 0011, 1110, 0111, 1111 and the longest 0-runs contribute 4+2+1+2+1+1+0 = 11.

%Y Cf. A005251, A119706, A333394, A333395.

%K nonn

%O 1,2

%A _Steven Finch_, Mar 18 2020