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.

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, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66

Offset

0,3

Comments

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));

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));

the table provides a permutation of the natural numbers when seen as flattened sequence.

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

Links

Reinhard Zumkeller, Rows n = 0..78 of triangle, flattened - all terms < 2^12

Reinhard Zumkeller, Illustration of initial terms

Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^21-1.

Michael De Vlieger, Fan style binary tree of a(n), 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.

Index entries for sequences related to binary expansion of n

Index entries for triangles and arrays related to Pascal's triangle

Index entries for sequences that are permutations of the nonnegative integers

Formula

A344085(n) = A019565(a(n-1)). - Michael De Vlieger, Jan 03 2025

Example

See link.

Mathematica

{{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jan 03 2025 *)

Prog

(Haskell)
import List (elemIndices)
a187769 n k = a187769_tabf !! n !! k
a187769_row n = a187769_tabf !! n
a187769_tabf = [0] : [elemIndices (b, len - b) $
   takeWhile ((<= len) . uncurry (+)) $ zip a000120_list a023416_list |
   len <- [1 ..], b <- [1 .. len]]
a187769_list = concat a187769_tabf

Crossrefs

Rows of A187786, duplicates removed;

Cf. A099627 (left edge), A023758 (right edge).

Cf. A019565, A294648, A344085 (A019565(a(n))).

Sequence in context: A239086 A306866 A379770 * A357492 A209862 A357491

Adjacent sequences: A187766 A187767 A187768 * A187770 A187771 A187772

Keyword

nonn,base,tabf

Author

Reinhard Zumkeller, Jan 05 2013

Status

approved