A333394 -id:A333394 - cp's OEIS frontend

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

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) = 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.

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.

Cf. A005251, A119706, A333394, A333396.

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 *)

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)).

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

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

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