cp's OEIS Frontend

A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.

%I A187769 #27 Feb 26 2025 20:34:18
%S A187769 0,1,2,3,4,5,6,7,8,9,10,12,11,13,14,15,16,17,18,20,24,19,21,22,25,26,
%T A187769 28,23,27,29,30,31,32,33,34,36,40,48,35,37,38,41,42,44,49,50,52,56,39,
%U A187769 43,45,46,51,53,54,57,58,60,47,55,59,61,62,63,64,65,66
%N A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.
%C A187769 Row lengths are given by Pascal's triangle (cf. A007318), seen as flattened sequence, or for n > 0: length of n-th row = A007318(A003056(n-1),A002262(n-1));
%C A187769 1 <= i < j <= length of n-th row: A023416(T(n,i)) = A023416(T(n,j)), A000120(T(n,i)) = A000120(T(n,j)) and A070939(T(n,i)) = A070939(T(n,j));
%C A187769 the table provides a permutation of the natural numbers when seen as flattened sequence.
%C A187769 This sequence can be seen as an irregular triangle S(i,k) where row 0 = {1}, row n = { m = 2^(n-1)..2^n - 1 } sorted according to omega(A019565(m)), where omega = A001221. Under this arrangement, the rows can be further subdivided into segments of m with the same omega(m), which align with the original definition's triangle T. - _Michael De Vlieger_, Jan 03 2025
%H A187769 Reinhard Zumkeller, <a href="/A187769/b187769.txt">Rows n = 0..78 of triangle, flattened - all terms < 2^12</a>
%H A187769 Reinhard Zumkeller, <a href="/A187769/a187769.txt">Illustration of initial terms</a>
%H A187769 Michael De Vlieger, <a href="/A187769/a187769.png">Log log scatterplot of a(n)</a>, n = 0..2^21-1.
%H A187769 Michael De Vlieger, <a href="/A187769/a187769_1.png">Fan style binary tree of a(n)</a>, n = 0..8192, i.e., rows 0..12, with a color function associated with (a(n) mod 2) / 2^floor(log_2 n) that illustrates the relationship with Pascal's triangle.
%H A187769 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A187769 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%H A187769 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the nonnegative integers</a>
%e A187769 See link.
%t A187769 {{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten (* _Michael De Vlieger_, Jan 03 2025 *)
%o A187769 (Haskell)
%o A187769 import List (elemIndices)
%o A187769 a187769 n k = a187769_tabf !! n !! k
%o A187769 a187769_row n = a187769_tabf !! n
%o A187769 a187769_tabf = [0] : [elemIndices (b, len - b) $
%o A187769    takeWhile ((<= len) . uncurry (+)) $ zip a000120_list a023416_list |
%o A187769    len <- [1 ..], b <- [1 .. len]]
%o A187769 a187769_list = concat a187769_tabf
%Y A187769 Rows of A187786, duplicates removed;
%Y A187769 Cf. A099627 (left edge), A023758 (right edge).
%Y A187769 Cf. A019565, A294648, A344084.
%K A187769 nonn,base,tabf
%O A187769 0,3
%A A187769 _Reinhard Zumkeller_, Jan 05 2013