A330036 The length of the largest run of 0's in the binary expansion of n + the length of the largest run of 1's in the binary expansion of n.

1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 2, 3, 4, 3, 4, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 5, 4, 5, 5, 6, 5, 4, 5, 3, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 6, 5, 4, 4, 4, 3, 3, 4, 6, 5, 4, 4, 6, 5, 6, 6, 7, 6, 5, 6, 4, 4, 5, 6, 4, 3, 3, 4, 4, 4, 5, 6, 5, 4, 3

Offset

0,3

Comments

All numbers appear in this sequence. The number of 1's in the n-th Mersenne number (A000225) is n and the number of 0's in the n-th Mersenne number is 0. 0+n=n. See formula.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

a(n) = A087117(n) + A038374(n).

a(A000225(n)) = n for n > 0.

Example

   n [binary n ]  A087117(n) + A038374(n) = a(n)
   0 [       0 ]  1          + 0          = 1
   1 [       1 ]  0          + 1          = 1
   2 [     1 0 ]  1          + 1          = 2
   3 [     1 1 ]  0          + 2          = 2
   4 [   1 0 0 ]  2          + 1          = 3
   5 [   1 0 1 ]  1          + 1          = 2
   6 [   1 1 0 ]  1          + 2          = 3
   7 [   1 1 1 ]  0          + 3          = 3
   8 [ 1 0 0 0 ]  3          + 1          = 4
   9 [ 1 0 0 1 ]  2          + 1          = 3
  10 [ 1 0 1 0 ]  1          + 1          = 2
  11 [ 1 0 1 1 ]  1          + 2          = 3
  12 [ 1 1 0 0 ]  2          + 2          = 4
  13 [ 1 1 0 1 ]  1          + 2          = 3
  14 [ 1 1 1 0 ]  1          + 3          = 4
  15 [ 1 1 1 1 ]  0          + 4          = 4

Maple

f:= proc(n) local L;
  L:= convert(n, base, 2);
  max(map(nops, [ListTools:-Split(`=`, L, 1)]))+max(map(nops, [ListTools:-Split(`=`, L, 0)]))
end proc:
map(f, [$0..100]); # Robert Israel, Apr 06 2020

Mathematica

Table[Sum[Max[Differences[Position[Flatten@{k, IntegerDigits[n, 2], k}, k]]], {k, 0, 1}]-2, {n, 0, 82}]

Crossrefs

Cf. A000120, A000225, A007088, A023416, A038374, A087117.

Sequence in context: A182745 A129843 A349043 * A050430 A277329 A071330

Adjacent sequences: A330033 A330034 A330035 * A330037 A330038 A330039

Keyword

nonn,base

Author

Joshua Oliver, Nov 27 2019

Status

approved