A359400 Sum of positions of zeros in the reversed binary expansion of n, where positions in a sequence are read starting with 1 from the left.

1, 0, 1, 0, 3, 2, 1, 0, 6, 5, 4, 3, 3, 2, 1, 0, 10, 9, 8, 7, 7, 6, 5, 4, 6, 5, 4, 3, 3, 2, 1, 0, 15, 14, 13, 12, 12, 11, 10, 9, 11, 10, 9, 8, 8, 7, 6, 5, 10, 9, 8, 7, 7, 6, 5, 4, 6, 5, 4, 3, 3, 2, 1, 0, 21, 20, 19, 18, 18, 17, 16, 15, 17, 16, 15, 14, 14, 13

Offset

0,5

Links

Table of n, a(n) for n=0..77.

Formula

a(n) = binomial(A029837(n)+1, 2) - A029931(n), for n>0.

Example

The reversed binary expansion of 100 is (0,0,1,0,0,1,1), with zeros at positions {1,2,4,5}, so a(100) = 12.

Mathematica

Table[Total[Join@@Position[Reverse[IntegerDigits[n, 2]], 0]], {n, 0, 100}]

Prog

(C)
long A359400(long n) {
  long result = 0, counter = 1;
  do {
    if (n % 2 == 0)
      result += counter;
    counter++;
    n /= 2;
  } while (n > 0);
  return result; } // Frank Hollstein, Jan 06 2023
(Python)
def a(n): return sum(i for i, bi in enumerate(bin(n)[:1:-1], 1) if bi=='0')
print([a(n) for n in range(78)]) # Michael S. Branicky, Jan 09 2023

Crossrefs

The number of zeros is A023416, partial sums A059015.

Row sums of A368494.

For positions of 1's we have A029931, non-reversed A230877.

The non-reversed version is A359359.

A003714 lists numbers with no successive binary indices.

A030190 gives binary expansion, reverse A030308.

A039004 lists the positions of zeros in A345927.

Cf. A000120, A048793, A069010, A070939, A073642, A328594, A328595, A344618, A359402, A359495.

Sequence in context: A194763 A194741 A194753 * A373072 A323908 A098825

Adjacent sequences: A359397 A359398 A359399 * A359401 A359402 A359403

Keyword

nonn,base

Author

Gus Wiseman, Jan 05 2023

Status

approved