OFFSET
0,3
COMMENTS
This is a permutation of the natural numbers; A160679 is the inverse permutation. - Jianing Song, Aug 10 2022
REFERENCES
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..1000
G. P. Michon, Discussion of Conway's On2 [From John W. Layman, Nov 05 2010]
FORMULA
a(n) = A051775(n,n).
From Jianing Song, Aug 10 2022: (Start)
If n = Sum_j 2^e(j), then a(n) is the XOR of A006017(e(j))'s. Proof: let N+ = XOR and N* denote the nim addition and the nim multiplication, then n N* n = (Sum_j 2^e(j)) N* (Sum_j 2^e(j)) = (Nim-sum_j 2^e(j)) N* (Nim-sum_j 2^e(j)) = (Nim-sum_j (2^e(j) N* 2^e(j))) N+ (Nim-sum_{i<j} ((2^e(i) N* 2^e(j)) N+ (2^e(j) N* 2^e(i)))) = (Nim-sum_j (2^e(j) N* 2^e(j))) N+ (Nim-sum_{i<j} 0) = Nim-sum_j (2^e(j) N* 2^e(j)).
For example, for n = 11 = 2^0 + 2^1 + 2^3, a(11) = A006017(0) XOR A006017(1) XOR A006017(3) = 1 XOR 3 XOR 13 = 15.
More generally, if n = Sum_j 2^e(j), k is a power of 2, then the nim k-th power of n is the XOR of (nim k-th power of 2^e(j))'s. (End)
MAPLE
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
a(1)-a(49) confirmed, a(50)-a(71) added by John W. Layman, Nov 05 2010
a(0) prepended by Jianing Song, Aug 10 2022
STATUS
approved