%I #107 Nov 20 2024 08:39:22

%S 2,9,10,12,35,37,38,41,42,44,49,50,52,56,135,139,141,142,147,149,150,

%T 153,154,156,163,165,166,169,170,172,177,178,180,184,195,197,198,201,

%U 202,204,209,210,212,216,225,226,228,232,240,527,535,539,541,542,551

%N Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.

%C Also numbers k such that the binary digital mean dm(2, k) = (Sum_{i=1..d} 2*d_i - 1) / (2*d) = 0, where d is the number of digits in the binary representation of k and d_i the individual digits. - _Reikku Kulon_, Sep 21 2008

%C From _Reikku Kulon_, Sep 29 2008: (Start)

%C Each run of values begins with 2^(2k + 1) + 2^(k + 1) - 2^k - 1. The initial values increase according to the sequence {2^(k - 1), 2^(k - 2), 2^(k - 3), ..., 2^(k - k)}.

%C After this, the values follow a periodic sequence of increases by successive powers of two with single odd values interspersed.

%C Each run ends with an odd increase followed by increases of {2^(k - k), ..., 2^(k - 2), 2^(k - 1), 2^k}, finally reaching 2^(2k + 2) - 2^(k + 1).

%C Similar behavior occurs in other bases. (End)

%C Numbers k such that A000120(k)/A070939(k) = 1/2. - _Ctibor O. Zizka_, Oct 15 2008

%C Subsequence of A053754; A179888 is a subsequence. - _Reinhard Zumkeller_, Jul 31 2010

%C A000120(a(n)) = A023416(a(n)); A037861(a(n)) = 0.

%C A001700 gives number of terms having length 2*n in binary representation: A001700(n-1) = #{m: A070939(a(m))=2*n}. - _Reinhard Zumkeller_, Jun 08 2011

%C The number of terms below 2^k is A079309(floor(k/2)) for k > 1. - _Amiram Eldar_, Nov 21 2020

%H Reikku Kulon, <a href="/A031443/b031443.txt">Table of n, a(n) for n = 1..10000</a>

%H Jason Bell, Thomas F. Lidbetter and Jeffrey Shallit, <a href="https://doi.org/10.1142/S0129054120410014">Additive number theory via approximation by regular languages</a>, International Journal of Foundations of Computer Science, Vol. 31, No. 6 (2020), pp. 667-687; <a href="https://arxiv.org/abs/1804.07996">arXiv preprint</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 Reinhard Zumkeller, <a href="/A031443/a031443.hs.txt">Haskell Programs for Binary Digitally Balanced Numbers</a>.

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

%F a(n+1) = a(n) + 2^k + 2^(m-1) - 1 + floor((2^(k+m) - 2^k)/a(n))*(2^(2*m) + 2^(m-1)) where k is the largest integer such that 2^k divides a(n) and m is the largest integer such that 2^m divides a(n)/2^k+1. - Ulrich Schimke (UlrSchimke(AT)aol.com)

%F A145037(a(n)) = 0. - _Reikku Kulon_, Oct 02 2008

%e 9 is a term because '1001' contains 2 '0's and 2 '1's.

%p a:=proc(n) local nn, n1, n0: nn:=convert(n,base,2): n1:=add(nn[i],i=1..nops(nn)): n0:=nops(nn)-n1: if n0=n1 then n else end if end proc: seq(a(n), n = 1..240); # _Emeric Deutsch_, Jul 31 2008

%t Select[Range[250],DigitCount[#,2,1]==DigitCount[#,2,0]&] (* _Harvey P. Dale_, Jul 22 2013 *)

%t FromDigits[#,2]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},2n,{1,0}],{n,5}],1],_?(#[[1]]==0&)]//Sort (* _Harvey P. Dale_, May 30 2016 *)

%o (PARI) for(n=1,100,b=binary(n); l=length(b); if(sum(i=1,l, component(b,i))==l/2,print1(n,",")))

%o (PARI) is(n)=hammingweight(n)==hammingweight(bitneg(n,#binary(n))) \\ _Charles R Greathouse IV_, Mar 29 2013

%o (PARI) is(n)=2*hammingweight(n)==exponent(n)+1 \\ _Charles R Greathouse IV_, Apr 18 2020

%o (Magma) [ n: n in [2..250] | Multiplicity({* z: z in Intseq(n,2) *}, 0) eq &+Intseq(n,2) ]; // _Bruno Berselli_, Jun 07 2011

%o (Haskell) -- See link, showing that Ulrich Schimkes formula provides a very efficient algorithm. _Reinhard Zumkeller_, Jun 15 2011

%o (Perl) for my $half ( 1 .. 4 ) {

%o my $N = 2 * $half; # only even widths apply

%o my $vector = (1 << ($N-1)) | ((1 << ($N/2-1)) - 1); # first key

%o my $n = 1; $n *= $_ for 2 .. $N; # N!

%o my $d = 1; $d *= $_ for 2 .. $N/2; # (N/2)!

%o for (1 .. $n/($d*$d*2)) {

%o print "$vector, ";

%o my ($v, $d) = ($vector, 0);

%o until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }

%o $vector += $d + 1 + (($v ^ ($v + 1)) >> 2); # next key

%o }

%o } # _Ruud H.G. van Tol_, Mar 30 2014

%o (Python)

%o from sympy.utilities.iterables import multiset_permutations

%o A031443_list = [int('1'+''.join(p),2) for n in range(1,10) for p in multiset_permutations('0'*n+'1'*(n-1))] # _Chai Wah Wu_, Nov 15 2019

%Y Subsequences: A144798, A144799, A144800, A144801, A144812, A179888.

%Y Subsequence of A053754.

%Y Row n = 2 of A378000.

%Y Cf. A049354-A049360, A000120, A001316, A006519, A023416, A070939, A144777, A145057, A145058, A145059, A145060, A144912, A145037, A191292, A090050, A014486, A061854, A037861, A079309.

%Y Terms of binary width n are enumerated by A001700.

%K nonn,base,easy,nice,changed

%O 1,1

%A _Clark Kimberling_