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A081706
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Numbers n such that binary representation ends either in an odd number of ones followed by one zero or in an even number of ones.
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22
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2, 3, 10, 11, 14, 15, 18, 19, 26, 27, 34, 35, 42, 43, 46, 47, 50, 51, 58, 59, 62, 63, 66, 67, 74, 75, 78, 79, 82, 83, 90, 91, 98, 99, 106, 107, 110, 111, 114, 115, 122, 123, 130, 131, 138, 139, 142, 143, 146, 147, 154, 155, 162, 163, 170, 171, 174, 175, 178, 179, 186
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OFFSET
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1,1
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COMMENTS
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Values of k such that the Motzkin number A001006(k) is even. Values of k such that the number of restricted hexagonal polyominoes with k+1 cells (A002212) is even.
Or union of sequences {2*A079523(n)+k}, k=0,1. A generalization see in comment to A161639. - Vladimir Shevelev, Jun 15 2009
Or intersection of sequences A121539 and {A121539(n)-1}. A generalization see in comment to A161890. - Vladimir Shevelev, Jul 03 2009
Also numbers n for which A010060(n+2) = A010060(n). - Vladimir Shevelev, Jul 06 2009
The asymptotic density of this sequence is 1/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021
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LINKS
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Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
Jean-Paul Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 2 (2015), pp. 375-388; arXiv preprint, arXiv:1401.3727 [math.NT], 2014.
Jean-Paul Allouche, André Arnold, Jean Berstel, Srećko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., Vol. 139, No. 1-3 (1995), pp. 455-461.
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.
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FORMULA
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a(2n-1) = 2*A079523(n) = 4*A003159(n)-2; a(2n) = 4*A003159(n)-1.
Note that a(2n) = 1+a(2n-1).
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MATHEMATICA
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(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[Range[200], Mod[m[#], 2] == 0 &] (* Jean-François Alcover, Jul 10 2013 *)
Select[Range[200], EvenQ@Hypergeometric2F1[3/2, -#, 3, 4]&] (* Vladimir Reshetnikov, Nov 02 2015 *)
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PROG
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(PARI) is(n)=valuation(bitor(n, 1)+1, 2)%2==0 \\ Charles R Greathouse IV, Mar 07 2013
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CROSSREFS
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Cf. A001006, A002212, A003159, A010060, A079523, A121539, A161639, A161890.
Sequence in context: A278742 A250174 A285622 * A032804 A248407 A047473
Adjacent sequences: A081703 A081704 A081705 * A081707 A081708 A081709
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KEYWORD
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nonn,base,easy
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AUTHOR
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Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
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STATUS
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approved
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