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A079523
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Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.
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35
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1, 5, 7, 9, 13, 17, 21, 23, 25, 29, 31, 33, 37, 39, 41, 45, 49, 53, 55, 57, 61, 65, 69, 71, 73, 77, 81, 85, 87, 89, 93, 95, 97, 101, 103, 105, 109, 113, 117, 119, 121, 125, 127, 129, 133, 135, 137, 141, 145, 149, 151, 153, 157, 159, 161, 165, 167, 169, 173, 177, 181
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OFFSET
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1,2
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COMMENTS
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Sequence of n such that a(n) = 3n begins 7, 23, 27, 29, 31, 39, 71, 87, 91, 93, 95, ...
Values of k such that the Motzkin number A001006(2k) is even. Values of k such that the number of restricted hexagonal polyominoes with 2k+1 cells is even (see A002212). Values of k such that the number of directed animals of size k+1 is even (see A005773). Values of k such that the Riordan number A005043(k) is even. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
The sequence has the following fractal property: remove terms of the form 4k+1 from the sequence, and the remaining terms are of the form 4k+3: 7, 23, 31, 39, 55, 71, 87, ...; then subtract 3 from each of these terms and divide by 4 and you get the original sequence: 1, 5, 7, 9, 13, ... - Benoit Cloitre, Apr 06 2010
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LINKS
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J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, Simon Plouffe, and B. E. Sagan, A relative of the Thue-Morse sequence, Discrete Math., 139, 1995, 455-461.
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FORMULA
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a(n) is asymptotic to 3n.
a(n) = 2*A003159(n) - 1. a(1)=1, a(n) = a(n-1) + 2 if (a(n-1)+1)/2 does not belong to the sequence and a(n) = a(n-1) + 4 otherwise. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
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MATHEMATICA
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Select[ Range[200], MatchQ[ IntegerDigits[#, 2], {b : (1) ..} | {___, 0, b : (1) ..} /; OddQ[ Length[{b}]]] & ] (* Jean-François Alcover, Jun 17 2013 *)
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PROG
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(Haskell)
import Data.List (elemIndices)
a079523 n = a079523_list !! (n-1)
a079523_list = elemIndices 0 a035263_list
(Magma) [n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 + 1]; // Vincenzo Librandi, Apr 16 2015
(Python)
from itertools import count, islice
def A079523_gen(startvalue=1): return filter(lambda n:(~(n+1)&n).bit_length()&1, count(max(startvalue, 1))) # generator of terms >= startvalue
A079523_list = list(islice(A079523_gen(), 30)) # Chai Wah Wu, Jul 05 2022
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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