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A382467 Irregular triangle read by rows, where row n lists the integers from 0 to 2^n - 1 sorted by the number of zeros in their binary representation (in case of ties, by their decimal value).

%I A382467 #16 Apr 02 2025 18:48:40
%S A382467 0,0,1,0,1,3,2,0,1,3,7,2,5,6,4,0,1,3,7,15,2,5,6,11,13,14,4,9,10,12,8,
%T A382467 0,1,3,7,15,31,2,5,6,11,13,14,23,27,29,30,4,9,10,12,19,21,22,25,26,28,
%U A382467 8,17,18,20,24,16,0,1,3,7,15,31,63,2,5,6,11,13,14,23,27,29,30
%N A382467 Irregular triangle read by rows, where row n lists the integers from 0 to 2^n - 1 sorted by the number of zeros in their binary representation (in case of ties, by their decimal value).
%H A382467 Paolo Xausa, <a href="/A382467/b382467.txt">Table of n, a(n) for n = 0..8190</a> (rows 0..12 of triangle, flattened).
%e A382467 Triangle begins:
%e A382467   [0]  0;
%e A382467   [1]  0,  1;
%e A382467   [2]  0,  1,  3,  2;
%e A382467   [3]  0,  1,  3,  7,  2,  5,  6,  4;
%e A382467   [4]  0,  1,  3,  7, 15,  2,  5,  6, 11, 13, 14,  4,  9, 10, 12,  8;
%e A382467   ...
%p A382467 b:= proc(n) b(n):= add(1-i, i=Bits[Split](n)) end:
%p A382467 T:= n-> sort([$0..2^n-1], (x, y)-> b(x)<b(y) or b(x)=b(y) and x<y)[]:
%p A382467 seq(T(n), n=0..6);  # _Alois P. Heinz_, Mar 31 2025
%t A382467 Table[SortBy[Range[0, 2^n - 1], DigitCount[#, 2, 0] &], {n, 0, 6}]
%Y A382467 Cf. A080791, A294648 (sorted by number of ones).
%Y A382467 Main diagonal gives A000225.
%Y A382467 Last elements of rows give A131577.
%Y A382467 Row sums give A006516.
%K A382467 nonn,tabf,look,base,easy
%O A382467 0,6
%A A382467 _Paolo Xausa_, Mar 31 2025