|
%I #55 Jan 09 2023 14:52:04
%S 2,3,10,11,14,15,18,19,26,27,34,35,42,43,46,47,50,51,58,59,62,63,66,
%T 67,74,75,78,79,82,83,90,91,98,99,106,107,110,111,114,115,122,123,130,
%U 131,138,139,142,143,146,147,154,155,162,163,170,171,174,175,178,179,186
%N 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.
%C 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.
%C Or union of sequences {2*A079523(n)+k}, k=0,1. A generalization see in comment to A161639. - _Vladimir Shevelev_, Jun 15 2009
%C Or intersection of sequences A121539 and {A121539(n)-1}. A generalization see in comment to A161890. - _Vladimir Shevelev_, Jul 03 2009
%C Also numbers n for which A010060(n+2) = A010060(n). - _Vladimir Shevelev_, Jul 06 2009
%C The asymptotic density of this sequence is 1/3 (Rowland and Yassawi, 2015; Burns, 2016). - _Amiram Eldar_, Jan 30 2021
%C Numbers of the form 4^k*(2*n-1)-2 and 4^k*(2*n-1)-1 where n and k are positive integers. - _Michael Somos_, Oct 22 2021
%H Amiram Eldar, <a href="/A081706/b081706.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from G. C. Greubel)
%H Jean-Paul Allouche, <a href="https://jtnb.centre-mersenne.org/item/?id=JTNB_2015__27_2_375_0">Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence</a>, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 2 (2015), pp. 375-388; <a href="https://arxiv.org/abs/1401.3727">arXiv preprint</a>, arXiv:1401.3727 [math.NT], 2014.
%H Jean-Paul Allouche, André Arnold, Jean Berstel, Srećko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, <a href="http://dx.doi.org/10.1016/0012-365X(93)00147-W">A sequence related to that of Thue-Morse</a>, Discrete Math., Vol. 139, No. 1-3 (1995), pp. 455-461.
%H Rob Burns, <a href="https://arxiv.org/abs/1611.04910">Asymptotic density of Motzkin numbers modulo small primes</a>, arXiv:1611.04910 [math.NT], 2016.
%H Eric Rowland and Reem Yassawi, <a href="http://www.numdam.org/item/JTNB_2015__27_1_245_0/">Automatic congruences for diagonals of rational functions</a>, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.
%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%F a(2n-1) = 2*A079523(n) = 4*A003159(n)-2; a(2n) = 4*A003159(n)-1.
%F Note that a(2n) = 1+a(2n-1).
%t (* 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 *)
%t Select[Range[200], EvenQ@Hypergeometric2F1[3/2, -#, 3, 4]&] (* _Vladimir Reshetnikov_, Nov 02 2015 *)
%o (PARI) is(n)=valuation(bitor(n,1)+1,2)%2==0 \\ _Charles R Greathouse IV_, Mar 07 2013
%o (Python)
%o from itertools import count, islice
%o def A081706_gen(): # generator of terms
%o for n in count(0):
%o if (n&-n).bit_length()&1:
%o m = n<<2
%o yield m-2
%o yield m-1
%o A081706_list = list(islice(A081706_gen(),30)) # _Chai Wah Wu_, Jan 09 2023
%Y Cf. A001006, A002212, A003159, A010060, A079523, A121539, A161639, A161890.
%K nonn,base,easy
%O 1,1
%A _Emeric Deutsch_ and _Bruce E. Sagan_, Apr 02 2003
|
