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%I #41 Apr 18 2020 00:58:06
%S 1,3,5,6,7,11,13,14,15,19,21,22,23,25,26,27,28,29,30,31,39,43,45,46,
%T 47,51,53,54,55,57,58,59,60,61,62,63,71,75,77,78,79,83,85,86,87,89,90,
%U 91,92,93,94,95,99,101,102,103,105,106,107,108,109,110,111,113,114,115
%N Numbers which in base 2 have fewer 0's than 1's.
%C A037861(a(n)) < 0.
%C b_k = {a(n) | for all n s.t. a(n) contains k binary digits equal to 1} is the list of all valid win/loss round sequences in a "best of 2k-1" two player game, where 1 is a win and 0 is a loss. For example 19 = 10011b represents a game where the winner won the first two rounds, lost the next two, and won the last one. |b_k| = A001700(k). - _Philippe Beaudoin_, May 14 2014
%H T. D. Noe, <a href="/A072600/b072600.txt">Table of n, a(n) for n = 1..4733</a> ( numbers < 2^13)
%H Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, <a href="https://arxiv.org/abs/1804.07996">Additive Number Theory via Approximation by Regular Languages</a>, arXiv:1804.07996 [cs.FL], 2018.
%H Thomas Finn Lidbetter, <a href="https://uwspace.uwaterloo.ca/bitstream/handle/10012/14254/Lidbetter_Thomas.pdf">Counting, Adding, and Regular Languages</a>, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e 11 is present because '1011' contains 1 '0' and 3 '1's: 1<3.
%t Select[Range[130],DigitCount[#,2,0]<DigitCount[#,2,1]&] (* _Harvey P. Dale_, Jan 12 2011 *)
%o (Haskell)
%o a072600 n = a072600_list !! (n-1)
%o a072600_list = filter ((< 0) . a037861) [0..]
%o -- _Reinhard Zumkeller_, Mar 31 2015
%o (PARI) is(n)=2*hammingweight(n)>exponent(n)+1 \\ _Charles R Greathouse IV_, Apr 18 2020
%Y Cf. A007088, A000120, A023416, A072601, A031443, A072602, A072603, A037861
%K nonn,base,easy
%O 1,2
%A _Reinhard Zumkeller_, Jun 23 2002
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