A337821 - OEIS

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

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OFFSET

1,7

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 paraphrases (A003602).

LINKS

Table of n, a(n) for n=1..96.

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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.

Sequence in context: A304095 A276675 A236389 * A143078 A106405 A228601

Adjacent sequences: A337818 A337819 A337820 * A337822 A337823 A337824

KEYWORD

nonn,easy

AUTHOR

Peter Munn, Sep 23 2020

STATUS

approved