A187786 -id:A187786 - 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.

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

Reinhard Zumkeller, Jan 05 2013

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

			See link.

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

Rows of A187786, duplicates removed;
Cf. A099627 (left edge), A023758 (right edge).
Cf. A019565, A294648, A344084.

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

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

A361478 Irregular table T(n, k), n >= 0, k = 1..A361477(n), read by rows; the n-th row lists the integers whose binary expansions have the same multiset of run-lengths as that of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 4, 6, 7, 8, 14, 9, 11, 13, 10, 9, 11, 13, 12, 9, 11, 13, 8, 14, 15, 16, 30, 17, 23, 29, 18, 20, 22, 26, 19, 25, 27, 18, 20, 22, 26, 21, 18, 20, 22, 26, 17, 23, 29, 24, 28, 19, 25, 27, 18, 20, 22, 26, 19, 25, 27, 24, 28, 17, 23, 29, 16, 30

Offset: 0

Rémy Sigrist, Mar 13 2023

This sequence has similarities with A187786; here we consider multisets of run-lengths, there multisets of digits in binary expansions.

			Table T(n, k) begins (in decimal and in binary):
  n   n-th row    bin(n)  n-th row in binary
  --  ----------  ------  ------------------
   0  0                0  0
   1  1                1  1
   2  2               10  10
   3  3               11  11
   4  4, 6           100  100, 110
   5  5              101  101
   6  4, 6           110  100, 110
   7  7              111  111
   8  8, 14         1000  1000, 1110
   9  9, 11, 13     1001  1001, 1011, 1101
  10  10            1010  1010
  11  9, 11, 13     1011  1001, 1011, 1101
  12  12            1100  1100
  13  9, 11, 13     1101  1001, 1011, 1101
  14  8, 14         1110  1000, 1110
  15  15            1111  1111
  16  16, 30       10000  10000, 11110

Rémy Sigrist, Table of n, a(n) for n = 0..6009 (rows for n = 0..511 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A187786, A361477, A361479, A361480.

PARI
```
See Links section.
```

T(n, 1) = A361479(n).

T(n, A361477(n)) = A361480(n).

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.

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