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

Author

Joshua Oliver, Nov 27 2019

Keywords

• nonn
• base

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.

Examples

			   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
		

Links

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

Crossrefs

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

Programs

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}]

Formula

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

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