A333394 Total length of all longest runs of 0's in solus bitstrings of length n.

0, 1, 4, 9, 18, 34, 62, 110, 192, 331, 565, 958, 1615, 2710, 4531, 7552, 12554, 20823, 34472, 56972, 94020, 154959, 255102, 419532, 689312, 1131632, 1856382, 3043208, 4985674, 8163321, 13359207, 21851594, 35726470, 58386958, 95383471, 155766277, 254288786

Offset

0,3

Comments

A bitstring is solus if all of its 1's are isolated.

The number of these bitstrings is A000045(n+2).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.

Formula

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

Example

a(4) = 18 because the A000045(6) = 8 solus bitstrings of length 4 are 0000, 1000, 0100, 0010, 0001, 1010, 0101, 1001 and the longest 0-runs contribute 4+3+2+2+3+1+1+2 = 18.

Maple

b:= proc(n, w, m, s) option remember; `if`(n=0, m,
      b(n-1, w+1, max(m, w+1), irem(s, 10)*10+0)+
     `if`(s in [01, 21], 0, b(n-1, 0, m, irem(s, 10)*10+1)))
    end:
a:= n-> b(n, 0, 0, 22):
seq(a(n), n=0..39);  # Alois P. Heinz, Mar 18 2020

Mathematica

b[n_, w_, m_, s_] := b[n, w, m, s] = If[n == 0, m, b[n-1, w+1, Max[m, w+1], Mod[s, 10]*10+0]+If[MatchQ[s, 01|21], 0, b[n-1, 0, m, Mod[s, 10]*10+1]]];
a[n_] := b[n, 0, 0, 22];
a /@ Range[0, 39] (* Jean-François Alcover, Aug 21 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000045, A119706, A333395, A333396.

Sequence in context: A019453 A301104 A038089 * A301150 A301078 A320851

Adjacent sequences: A333391 A333392 A333393 * A333395 A333396 A333397

Keyword

nonn

Author

Steven Finch, Mar 18 2020

Status

approved