This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A294648 #71 Jul 28 2025 08:58:48
%S A294648 0,1,0,1,2,3,0,1,2,4,3,5,6,7,0,1,2,4,8,3,5,6,9,10,12,7,11,13,14,15,0,
%T A294648 1,2,4,8,16,3,5,6,9,10,12,17,18,20,24,7,11,13,14,19,21,22,25,26,28,15,
%U A294648 23,27,29,30,31,0,1,2,4,8,16,32,3,5,6,9,10,12,17,18,20,24,33,34,36,40,48,7,11,13,14,19,21
%N A294648 Irregular triangle read by rows, representing a family of sequences L(n), for n=1, 2, 3, ... The sequence L(n) (i.e., the n-th row) is the ordinance of vectors of the n-dimensional Boolean cube (hypercube) {0,1}^n in accordance with their (Hamming) weights, where the lexicographic order is chosen as a second criterion for an ordinance the vectors of equal weights.
%C A294648 The sequences represent the ordinance of vectors of the n-dimensional Boolean cube (hypercube) {0,1}^n in accordance with their (Hamming) weights. The lexicographic order is chosen as a second criterion for the ordinance of the vectors of equal weights. We refer to this order as a Weight-Lexicographic Order (WLO). The WLO is represented by the (serial) numbers of the vectors, instead of the vectors itself. It is well known that if the vectors of {0,1}^n are in lexicographic (standard) order, their numbers form the sequence of natural numbers 0, 1, 2, ..., 2^n-1. So, the WLO means a permutation of the numbers 0, 1, 2, ..., 2^n-1, such that the corresponding vectors are in WLO. This sequence (permutation) is denoted by L(n). It consists of (n+1) subsequences, corresponding to the layers of the Boolean cube.
%H A294648 Michael De Vlieger, <a href="/A294648/b294648.txt">Table of n, a(n) for n = 1..10000</a>
%H A294648 Valentin Bakoev, <a href="https://doi.org/10.1142/S179383092150021X">Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube</a>, Discrete Mathematics, Algorithms and Applications, Vol. 13 No 3, 2150021 (2021); <a href="https://arxiv.org/abs/1811.04421">arXiv preprint</a>, arXiv:1811.04421 [cs.DM], 2018-2020.
%H A294648 Valentin Bakoev, <a href="https://arxiv.org/abs/1905.08649">Fast Computing the Algebraic Degree of Boolean Functions</a>, arXiv:1905.08649 [cs.DM], 2019.
%F A294648 For n=1, 2, 3, ..., L(n) is defined by the recurrence:
%F A294648 if n=1, L(1)= 0, 1;
%F A294648 else L(n)= l(n, 0), l(n, 1), ..., l(n, k), ..., l(n, n), where the subsequences are defined as follows:
%F A294648 l(n, k)= 0, if k=0, else
%F A294648 l(n, k)= 2^n - 1, if k=n, else
%F A294648 l(n, k)= l(n-1, k), l(n-1, k-1) + 2^{n-1}, for 0 < k < n.
%F A294648 Comments:
%F A294648 1) l(n, k)= l(n-1, k), l(n-1, k-1) + 2^{n-1} means that l(n, k) is a concatenation of two subsequences: l(n-1, k) and l(n-1, k-1) + 2^{n-1}. The second one is obtained after addition of the number 2^{n-1} to each member of l(n-1,k-1).
%F A294648 2) The computing of the members of L(n) resembles the computing (and filling in) the binomial coefficients in Pascal's triangle. The binomial coefficients determine the lengths of the subsequences l(n, k), 0 <= k <= n, in L(n). Thus the beginning of each subsequence can be computed easily.
%F A294648 3) The inductive formula, corresponding to the recurrence, is much more useful for implementations.
%e A294648 The lexicographic order of {0,1}^3 is: (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1), and the corresponding sequence of (serial) numbers is: 0, 1, 2, ..., 7. The WLO of these vectors is given by the sequence L(3)= 0, 1, 2, 4, 3, 5, 6, 7.
%e A294648 The triangle starts:
%e A294648 n=1: 0, 1;
%e A294648 n=2: 0, 1, 2, 3;
%e A294648 n=3: 0, 1, 2, 4, 3, 5, 6, 7;
%e A294648 n=4: 0, 1, 2, 4, 8, 3, 5, 6, 9, 10, 12, 7, 11, 13, 14, 15;
%e A294648 n=5: 0, 1, 2, 4, 8, 16, 3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 15, 23, 27, 29, 30, 31;
%e A294648 n=6: 0, 1, 2, 4, 8, 16, 32, 3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 31, 47, 55, 59, 61, 62, 63;
%p A294648 with(ListTools): conc := (a,b,c) -> Flatten([op(a),[seq(op(j)+c, j in b)]], 1):
%p A294648 rec := proc(n,k) option remember; `if`(k=0, [0], `if`(k=n, [2^n-1], conc(rec(n-1,k), rec(n-1,k-1), 2^(n-1)))) end:
%p A294648 L := n -> `if`(n=1, [0,1], Flatten([seq(rec(n,k),k=0..n)], 1)):
%p A294648 Flatten([seq(L(n), n = 1..6)], 1); # _Peter Luschny_, Nov 06 2017
%t A294648 conc[a_List, b_List, c_] := Join[a, b + c];
%t A294648 rec[n_, k_] := rec[n, k] = If[k == 0, {0}, If[k == n, {2^n - 1}, conc[rec[n - 1, k], rec[n - 1, k - 1], 2^(n - 1)]]];
%t A294648 L[n_] := If[n == 1, {0, 1}, Flatten[Table[rec[n, k], {k, 0, n}]]];
%t A294648 Array[L, 6] // Flatten (* _Jean-François Alcover_, Jul 26 2018, after _Peter Luschny_ *)
%t A294648 Table[SortBy[Range[0, 2^n - 1], DigitSum[#, 2] &], {n, 6}] (* _Paolo Xausa_, Jul 28 2025 *)
%o A294648 (Python)
%o A294648 from itertools import product
%o A294648 def sortby(x): return (len(x), x.count('1'), x)
%o A294648 def agen(maxbindigs):
%o A294648 for i in range(1, maxbindigs+1):
%o A294648 for t in sorted([p for p in product("01", repeat=i)], key=sortby):
%o A294648 yield int("".join(t), 2)
%o A294648 print([an for an in agen(6)]) # _Michael S. Branicky_, Aug 13 2021
%o A294648 (PARI) cmph(x, y) = my(d=hammingweight(x)-hammingweight(y)); if (d, d, x-y);
%o A294648 row(n) = my(v=[0..2^n-1]); vecsort(v, cmph); \\ _Michel Marcus_, Sep 16 2023
%Y A294648 A051459 gives the orders' number of the vectors of {0,1}^n in accordance with their weights.
%Y A294648 Cf. A007318, A047869 (8th row), A091444 (as bits), A187769, A263327 (10th row).
%Y A294648 Cf. A382467.
%K A294648 nonn,tabf
%O A294648 1,5
%A A294648 _Valentin Bakoev_, Nov 06 2017