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For any positive integer n with binary digits (b_1, ..., b_w) (where b_1 = 1), the binary digits of a(n) are (b_1, b_3, ..., b_{2*ceiling(w/2)-1}); a(0) = 0.
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%I #33 Mar 28 2024 18:04:09

%S 0,1,1,1,2,3,2,3,2,2,3,3,2,2,3,3,4,5,4,5,6,7,6,7,4,5,4,5,6,7,6,7,4,4,

%T 5,5,4,4,5,5,6,6,7,7,6,6,7,7,4,4,5,5,4,4,5,5,6,6,7,7,6,6,7,7,8,9,8,9,

%U 10,11,10,11,8,9,8,9,10,11,10,11,12,13,12

%N For any positive integer n with binary digits (b_1, ..., b_w) (where b_1 = 1), the binary digits of a(n) are (b_1, b_3, ..., b_{2*ceiling(w/2)-1}); a(0) = 0.

%C In other words, we keep odd-indexed bits.

%C For any v > 0, the value v appears A003945(A070939(v)) times in the sequence.

%H Rémy Sigrist, <a href="/A371442/b371442.txt">Table of n, a(n) for n = 0..8192</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(A000695(n)) = n.

%F a(A001196(n)) = n.

%F a(A165199(n)) = a(n).

%e The first terms, in decimal and in binary, are:

%e n a(n) bin(n) bin(a(n))

%e -- ---- ------ ---------

%e 0 0 0 0

%e 1 1 1 1

%e 2 1 10 1

%e 3 1 11 1

%e 4 2 100 10

%e 5 3 101 11

%e 6 2 110 10

%e 7 3 111 11

%e 8 2 1000 10

%e 9 2 1001 10

%e 10 3 1010 11

%e 11 3 1011 11

%e 12 2 1100 10

%e 13 2 1101 10

%e 14 3 1110 11

%e 15 3 1111 11

%t A371442[n_] := FromDigits[IntegerDigits[n, 2][[1;;-1;;2]], 2];

%t Array[A371442, 100, 0] (* _Paolo Xausa_, Mar 28 2024 *)

%o (PARI) a(n) = { my (b = binary(n)); fromdigits(vector(ceil(#b/2), k, b[2*k-1]), 2); }

%o (Python) def a(n): return int(bin(n)[::2], 2)

%Y See A371459 for the sequence related to even-indexed bits.

%Y See A059905 and A063694 for similar sequences.

%Y Cf. A000695, A001196, A003945, A070939, A165199, A371461.

%K nonn,base,easy

%O 0,5

%A _Rémy Sigrist_, Mar 24 2024