A356028 Irregular triangle {A(n, k)} read by rows, giving in row n the numbers 1, 2, ..., 2^n - 1 ordered according to increasing binary weights, and for like weights decreasing.

1, 2, 1, 3, 4, 2, 1, 6, 5, 3, 7, 8, 4, 2, 1, 12, 10, 9, 6, 5, 3, 14, 13, 11, 7, 15, 16, 8, 4, 2, 1, 24, 20, 18, 17, 12, 10, 9, 6, 5, 3, 28, 26, 25, 22, 21, 19, 14, 13, 11, 7, 30, 29, 27, 23, 15, 31, 32, 16, 8, 4, 2, 1, 48, 40, 36, 34, 33, 24, 20, 18, 17, 12, 10, 9, 6, 5, 3, 56, 52, 50, 49, 44, 42, 41, 38, 37, 35, 28, 26, 25, 22, 21, 19, 14, 13, 11, 7, 60, 58, 57, 54, 53, 51, 46, 45, 43, 39, 30, 29, 27, 23, 15, 62, 61, 59, 55, 47, 31, 63

Offset

1,2

Comments

The length of row n is 2^n - 1 = A000225(n).

Also irregular triangle {A(n, k)} read by rows, giving in row n the numbers with a binary encoding of the list choose([n], m) = choose({1, 2,..., n}, m) (each encoding of length n), for n >= 1 and m = 1, 2, ..., n; written as entries for k = 1, 2, ..., 2^n - 1.

The binary encoding is obtained by setting 1s at the positions given by the choose([n],m) list of lists (in lexicographic order) and 0 otherwise. E.g., choose([3], 2) = [[1, 2], [1, 3], [2, 3]] with the encodings of length 3 [[1, 1, 0], [1, 0, 1], [0, 1, 1]], read as base 2 lists giving the numbers [6, 5, 3].

For the triangle T(n,m) of the sums of like m entries see A134346, (using offset 1).

The sum of row n gives A006516(n) = A171476(n-1), for n >= 1 (see A134346).

Links

Paolo Xausa, Table of n, a(n) for n = 1..8178 (rows 1..12 of the triangle, flattened).

Example

The irregular triangle A begins (commas separate the n subsequences for m = 1, 2, ..., n, corresponding to the binary encoded choose(n, m) lists or binary weights m):
n\k   1 2  3  4   5  6  7  8 9 10  11 12 13 14  15 ...
1:    1
2:    2 1, 3
3:    4 2  1, 6   5  3, 7
4:    8 4  2  1, 12 10  9  6 5  3, 14 13 11  7, 15
...
n = 5: [16 8 4 2 1, 24 20 18 17 12 10 9 6 5 3, 28 26 25 22 21 19 14 13 11 7, 30 29 27 23 15, 31];
n = 6: [32 16 8 4 2 1, 48 40 36 34 33 24 20 18 17 12 10 9 6 5 3, 56 52 50 49 44 42 41 38 37 35 28 26 25 22 21 19 14 13 11 7, 60 58 57 54 53 51 46 45 43 39 30 29 27 23 15, 62 61 59 55 47 31, 63);
...
A(4, 2) gives the number with the binary representation of the choose([4], 2) list [[1,1,0,0], [1,0,1,0], [1,0,0,1], [0,1,1,0], [0,1,0,1], [0,0,1,1]], obtained from the list choose([4], 2) = [[1,2], [1,3], [1,4], [2,3], [2,4], [3,4]], that is [12, 10, 9, 6, 5, 3].
A(4, 2) from the numbers 1, 2, ..., 15 with binary weight 2, that is of 3, 5, 6, 9, 10, 12, in decreasing order: 12, 10, 9, 6, 5, 3.

Mathematica

A356028row[n_]:=SortBy[Range[2^n-1], {DigitCount[#, 2, 1]&, -#&}];
Array[A356028row, 6] (* Paolo Xausa, Dec 20 2023 *)

Prog

(PARI) cmph(x, y) = my(d=hammingweight(x)-hammingweight(y)); if (d, d, y-x);
row(n) = my(v=[1..2^n-1]); vecsort(v, cmph); \\ Michel Marcus, Sep 16 2023

Crossrefs

Cf. A000225, A006516, A134346, A171476, A262881, A294648.

Sequence in context: A162598 A088208 A081878 * A088606 A140073 A333400

Adjacent sequences: A356025 A356026 A356027 * A356029 A356030 A356031

Keyword

nonn,tabf,base,look,easy

Author

Wolfdieter Lang, Jul 27 2022

Extensions

Name suggested by Kevin Ryde.

Status

approved