A356419 -id:A356419 - cp's OEIS frontend

A067576 Array T(i,j) read by downward antidiagonals, where T(i,j) is the j-th term whose binary expansion has i 1's.

1, 2, 3, 4, 5, 7, 8, 6, 11, 15, 16, 9, 13, 23, 31, 32, 10, 14, 27, 47, 63, 64, 12, 19, 29, 55, 95, 127, 128, 17, 21, 30, 59, 111, 191, 255, 256, 18, 22, 39, 61, 119, 223, 383, 511, 512, 20, 25, 43, 62, 123, 239, 447, 767, 1023, 1024, 24, 26, 45, 79, 125, 247, 479, 895, 1535, 2047

Offset: 1

Robert G. Wilson v, Jan 30 2002

This is a permutation of the positive integers; the inverse permutation is A356419. - Jianing Song, Aug 06 2022

			Array begins:
        j=1  j=2  j=3  j=4  j=5  j=6
  i=1:    1,   2,   4,   8,  16,  32, ...
  i=2:    3,   5,   6,   9,  10,  12, ...
  i=3:    7,  11,  13,  14,  19,  21, ...
  i=4:   15,  23,  27,  29,  30,  39, ...
  i=5:   31,  47,  55,  59,  61,  62, ...
  i=6:   63,  95, 111, 119, 123, 125, ...

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A356419.
T(n,n) gives A036563(n+1).
The antidiagonals are read in the opposite direction from those in A066884.
Antidiagonal sums give A361074.
Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691.

a = {}; Do[ a = Append[a, Last[ Take[ Select[ Range[2^13], Count[ IntegerDigits[ #, 2], 1] == j & ], i - j]]], {i, 2, 12}, {j, 1, i - 1} ]; a

A067587 Inverse of A066884 considered as a permutation of the positive integers.

Original entry on oeis.org

1, 3, 2, 6, 5, 9, 4, 10, 14, 20, 8, 27, 13, 19, 7, 15, 35, 44, 26, 54, 34, 43, 12, 65, 53, 64, 18, 76, 25, 33, 11, 21, 77, 90, 89, 104, 103, 118, 42, 119, 134, 151, 52, 169, 63, 75, 17, 135, 188, 208, 88, 229, 102, 117, 24, 251, 133, 150, 32, 168, 41, 51, 16, 28, 152

Offset: 1

Jared Ricks (jaredricks(AT)yahoo.com), Jan 31 2002

Cf. A066884, A000120, A068076, A263017, A356419.

w[n_] := Plus@@IntegerDigits[n, 2]; p[n_] := Plus@@MapThread[Binomial, {Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]]-1, Range[w[n]]}]; a[n_] := Binomial[w[n]+p[n], 2]+p[n]+1

a(n)=my(w=hammingweight(n), p=sum(i=1, n-1, hammingweight(i)==w)); binomial(w+p, 2) + p + 1 \\ Jianing Song, Aug 06 2022

foreach(1..10_000){$i=eval join "+", split //, sprintf "%b", $; $j=$r[$i]++; print "$ ",$j+1+($i+$j)*($i+$j-1)/2,"\n"} # Ivan Neretin, Mar 02 2016

from math import comb
def A067587(n):
    c, k = 0, 0
    for i,j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i,k)
    return comb(n.bit_count()+c,2)+c+1 # Chai Wah Wu, Mar 02 2023

Let w(n) = A000120(n) be the 'weight' of n; i.e. the number of 1's in the binary expansion of n. Let p(n) = A068076(n) be the number of positive integers < n with the same weight as n. Then a(n) = binomial(w(n)+p(n),2) + p(n) + 1.

Edited by Dean Hickerson, Feb 16 2002

Offset changed to 1 by Ivan Neretin, Mar 02 2016

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A067576 Array T(i,j) read by downward antidiagonals, where T(i,j) is the j-th term whose binary expansion has i 1's.

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A067587 Inverse of A066884 considered as a permutation of the positive integers.

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