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URL: https://oeis.org/A333395

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A333395
Total length of all longest runs of 1's in multus bitstrings of length n.
3
0, 2, 7, 16, 32, 62, 118, 221, 409, 751, 1371, 2492, 4513, 8148, 14674, 26371, 47304, 84717, 151508, 270622, 482849, 860661, 1532745, 2727483, 4849988, 8618549, 15306204, 27168300, 48199022, 85469639, 151495120, 268418323, 475405955, 841718780, 1489804565, 2636091495
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
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