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A303065 Number of numbers < n whose binary representation has the same difference between the numbers of 0's and 1's as n does.

%I A303065 #24 May 21 2025 12:21:44
%S A303065 0,0,0,0,1,1,2,0,0,1,2,1,3,2,3,0,0,2,3,3,4,4,5,1,5,6,7,2,8,3,4,0,0,1,
%T A303065 2,4,3,5,6,4,4,7,8,5,9,6,7,1,5,10,11,8,12,9,10,2,13,11,12,3,13,4,5,0,
%U A303065 0,1,2,6,3,7,8,9,4,9,10,10,11,11,12,5,5,12,13
%N A303065 Number of numbers < n whose binary representation has the same difference between the numbers of 0's and 1's as n does.
%C A303065 First occurrence of k, k=0,1,2,...: 0, 4, 6, 12, 20, 22, 25, 26, 28, 44, 49, ..., . - _Robert G. Wilson v_, Feb 08 2018
%C A303065 Ordinal transform of A037861, minus one. - _David Radcliffe_, May 21 2025
%H A303065 Alois P. Heinz, <a href="/A303065/b303065.txt">Table of n, a(n) for n = 0..32768</a>
%F A303065 a(n) = 0 iff n belongs to A097110. - _Rémy Sigrist_, May 16 2018
%e A303065 There are two numbers below 6 with number of 1's in the binary representation minus number of 0's equal to 1, namely 1 and 5, therefore a(6)=2.
%e A303065 There are 3 numbers below 12 with number of 1's in the binary representation minus number of 0's equal to 0, namely 2, 9, 10, therefore a(12)=3.
%p A303065 b:= n-> `if`(n=0, 1, add(1-2*i, i=Bits[Split](n))):
%p A303065 p:= proc() -1 end:
%p A303065 a:= proc(n) option remember; local t;
%p A303065       t:= b(n); p(t):= p(t)+1
%p A303065     end:
%p A303065 seq(a(n), n=0..82);  # _Alois P. Heinz_, May 21 2025
%t A303065 d[n_] := DigitCount[n, 2, 1] - DigitCount[n, 2, 0]; f[n_] := Block[{fd = d[n], c = k = 0}, While[k < n, If[d@ k == fd, c++]; k++]; c]; Array[f, 83, 0] (* _Robert G. Wilson v_, Feb 08 2018 *)
%o A303065 (Python)
%o A303065 d=[0]*200
%o A303065 for n in range(1024):
%o A303065     b = bin(n)[2:]
%o A303065     c0 = b.count('0')
%o A303065     c1 = len(b) - c0
%o A303065     diff = c0 - c1
%o A303065     print(d[100+diff], end=', ')
%o A303065     d[100+diff] += 1
%o A303065 (Python)
%o A303065 from collections import Counter
%o A303065 from itertools import count, islice
%o A303065 def a303065_gen():
%o A303065     counter = Counter()
%o A303065     for n in count():
%o A303065         bitstring = format(n, 'b')
%o A303065         diff = bitstring.count('1') - bitstring.count('0')
%o A303065         yield counter[diff]
%o A303065         counter[diff] += 1
%o A303065 a303065_list = list(islice(a303065_gen(), 83)) # _David Radcliffe_, May 21 2025
%Y A303065 Cf. A037861, A097110.
%K A303065 nonn,look,base
%O A303065 0,7
%A A303065 _Alex Ratushnyak_, Apr 17 2018
%E A303065 Offset corrected by _David Radcliffe_, May 21 2025