A356419 Inverse of A067576 considered as a permutation of the positive integers.

1, 2, 3, 4, 5, 8, 6, 7, 12, 17, 9, 23, 13, 18, 10, 11, 30, 38, 24, 47, 31, 39, 14, 57, 48, 58, 19, 69, 25, 32, 15, 16, 68, 80, 81, 93, 94, 108, 40, 107, 123, 139, 49, 156, 59, 70, 20, 122, 174, 193, 82, 213, 95, 109, 26, 234, 124, 140, 33, 157, 41, 50, 21, 22, 138, 155, 256

Offset: 1

Author

Jianing Song, Aug 06 2022

Keywords

• nonn
• look
• easy

Examples

			A067576(12) = 9, so a(9) = 12.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A067576, A000120, A068076, A263017, A067587.

Programs

PARI

a(n)=my(w=hammingweight(n), p=sum(i=1, n-1, hammingweight(i)==w)); binomial(w+p+1, 2) - p

Python

from math import comb
def A356419(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+1,2)-c # Chai Wah Wu, Mar 02 2023

Formula

Let w(n) = A000120(n) be the Hamming weight of n, p(n) = A068076(n), then a(n) = binomial(w(n)+p(n)+1, 2) - p(n).