%I #24 Sep 13 2024 06:54:05

%S 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,

%T 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,

%U 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

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

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

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

%C The 2-adic valuation of Kimberling's paraphrases (A003602).

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

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

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

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

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

%e Start of table showing the interleaving with ruler sequence, A007814:

%e n a(n) A007814 a(n/2)

%e ((n+1)/2)

%e 1 0 0

%e 2 0 0

%e 3 1 1

%e 4 0 0

%e 5 0 0

%e 6 1 1

%e 7 2 2

%e 8 0 0

%e 9 0 0

%e 10 0 0

%e 11 1 1

%e 12 1 1

%e 13 0 0

%e 14 2 2

%e 15 3 3

%e 16 0 0

%e 17 0 0

%e 18 0 0

%e 19 1 1

%e 20 0 0

%e 21 0 0

%e 22 1 1

%e 23 2 2

%e 24 1 1

%t a[n_] := IntegerExponent[(n/2^IntegerExponent[n, 2] + 1)/2, 2]; Array[a, 100] (* _Amiram Eldar_, Sep 30 2020 *)

%o (PARI) a(n) = valuation(n>>valuation(n,2)+1, 2) - 1; \\ _Kevin Ryde_, Apr 06 2024

%Y Odd bisection: A007814.

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

%Y Positions of zeros: A091072.

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

%K nonn,easy

%O 1,7

%A _Peter Munn_, Sep 23 2020