A337821 For n >= 0, a(4n+1) = 0, a(4n+3) = a(2n+1) + 1, a(2n+2) = a(n+1).

0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 0, 3, 4, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 3, 1, 0, 0, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 0, 4, 5, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 0, 3, 4, 1

Offset: 1

Author

Peter Munn, Sep 23 2020

Keywords

• nonn
• easy

Comments

This sequence is the ruler sequence A007814 interleaved with this sequence; specifically, the odd bisection is A007814, the even bisection is the sequence itself.

The 3-adic valuation of the Doudna sequence (A005940).

The 2-adic valuation of Kimberling's paraphrase (A003602) of the binary number system. [Edited Peter Munn, Aug 13 2025.]

Examples

			Start of table showing the interleaving with ruler sequence, A007814:
   n  a(n)  A007814    a(n/2)
            ((n+1)/2)
   1   0       0
   2   0                 0
   3   1       1
   4   0                 0
   5   0       0
   6   1                 1
   7   2       2
   8   0                 0
   9   0       0
  10   0                 0
  11   1       1
  12   1                 1
  13   0       0
  14   2                 2
  15   3       3
  16   0                 0
  17   0       0
  18   0                 0
  19   1       1
  20   0                 0
  21   0       0
  22   1                 1
  23   2       2
  24   1                 1
		

Crossrefs

Odd bisection: A007814.

A000265, A003602, A005940, A007949 are used in a formula defining this sequence.

Positions of zeros: A091072.

Sequences with similar interleaving: A089309, A014577, A025480, A034947, A038189, A082392, A099545, A181363, A274139.

Programs

Mathematica

a[n_] := IntegerExponent[(n/2^IntegerExponent[n, 2] + 1)/2, 2]; Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

PARI

a(n) = valuation(n>>valuation(n,2)+1, 2) - 1; \\ Kevin Ryde, Apr 06 2024

Formula

a(2*n) = a(n).

a(2*n+1) = A007814(n+1).

a(n) = A007949(A005940(n)).

a(n) = A007814(A003602(n)) = A007814((A000265(n)+1) / 2) = A089309(n) - 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Sep 13 2024