A333395 - OEIS

%I #34 Mar 24 2020 19:08:44

%S 0,2,7,16,32,62,118,221,409,751,1371,2492,4513,8148,14674,26371,47304,

%T 84717,151508,270622,482849,860661,1532745,2727483,4849988,8618549,

%U 15306204,27168300,48199022,85469639,151495120,268418323,475405955,841718780,1489804565,2636091495

%N Total length of all longest runs of 1'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 Alois P. Heinz, <a href="/A333395/b333395.txt">Table of n, a(n) for n = 1..985</a>

%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/((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)).

%e 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.

%t 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}];

%t ser[n_] := Series[gf[n], {x, 0, n}];

%t Drop[CoefficientList[ser[36], x], 1] (* _Peter Luschny_, Mar 19 2020 *)

%Y Cf. A005251, A119706, A333394, A333396.

%K nonn

%O 1,2

%A _Steven Finch_, Mar 18 2020