A333395 Total length of all longest runs of 1's in multus bitstrings of length n.

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

Alois P. Heinz, Table of n, a(n) for n = 1..985

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

Cf. A005251, A119706, A333394, A333396.

Sequence in context: A228189 A023550 A332232 * A130869 A212576 A208995

Adjacent sequences: A333392 A333393 A333394 * A333396 A333397 A333398

Keyword

nonn

Author

Steven Finch, Mar 18 2020

Status

approved