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%I A337821 #24 Sep 13 2024 06:54:05
%S A337821 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 A337821 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 A337821 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 A337821 For n >= 0, a(4n+1) = 0, a(4n+3) = a(2n+1) + 1, a(2n+2) = a(n+1).
%C A337821 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 A337821 The 3-adic valuation of the Doudna sequence (A005940).
%C A337821 The 2-adic valuation of Kimberling's paraphrases (A003602).
%F A337821 a(2*n) = a(n).
%F A337821 a(2*n+1) = A007814(n+1).
%F A337821 a(n) = A007949(A005940(n)).
%F A337821 a(n) = A007814(A003602(n)) = A007814((A000265(n)+1) / 2) = A089309(n) - 1.
%F A337821 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - _Amiram Eldar_, Sep 13 2024
%e A337821 Start of table showing the interleaving with ruler sequence, A007814:
%e A337821    n  a(n)  A007814    a(n/2)
%e A337821             ((n+1)/2)
%e A337821    1   0       0
%e A337821    2   0                 0
%e A337821    3   1       1
%e A337821    4   0                 0
%e A337821    5   0       0
%e A337821    6   1                 1
%e A337821    7   2       2
%e A337821    8   0                 0
%e A337821    9   0       0
%e A337821   10   0                 0
%e A337821   11   1       1
%e A337821   12   1                 1
%e A337821   13   0       0
%e A337821   14   2                 2
%e A337821   15   3       3
%e A337821   16   0                 0
%e A337821   17   0       0
%e A337821   18   0                 0
%e A337821   19   1       1
%e A337821   20   0                 0
%e A337821   21   0       0
%e A337821   22   1                 1
%e A337821   23   2       2
%e A337821   24   1                 1
%t A337821 a[n_] := IntegerExponent[(n/2^IntegerExponent[n, 2] + 1)/2, 2]; Array[a, 100] (* _Amiram Eldar_, Sep 30 2020 *)
%o A337821 (PARI) a(n) = valuation(n>>valuation(n,2)+1, 2) - 1; \\ _Kevin Ryde_, Apr 06 2024
%Y A337821 Odd bisection: A007814.
%Y A337821 A000265, A003602, A005940, A007949 are used in a formula defining this sequence.
%Y A337821 Positions of zeros: A091072.
%Y A337821 Sequences with similar interleaving: A089309, A014577, A025480, A034947, A038189, A082392, A099545, A181363, A274139.
%K A337821 nonn,easy
%O A337821 1,7
%A A337821 _Peter Munn_, Sep 23 2020

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