A333395 -id:A333395 - cp's OEIS frontend

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

Steven Finch, Mar 18 2020

nonn

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

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

			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.

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.

Cf. A000045, A119706, A333395, A333396.

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

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 *)

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)).

A333396 Total length of all longest runs of 0's in multus bitstrings of length n.

Original entry on oeis.org

1, 2, 5, 11, 23, 45, 87, 165, 309, 573, 1056, 1934, 3527, 6408, 11605, 20960, 37771, 67928, 121949, 218595, 391302, 699610, 1249475, 2229329, 3974083, 7078658, 12599318, 22410548, 39837420, 70775727, 125675525, 223052519, 395702395, 701695820, 1243827018, 2204007329

Offset: 1

Steven Finch, Mar 18 2020

nonn

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).

			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.

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

Cf. A005251, A119706, A333394, A333395.

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)).

cp's OEIS Frontend

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

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