OFFSET
0,2
COMMENTS
From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 2,3,1.
This is a permutation of the natural numbers; A004468 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)
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
Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
FORMULA
From Jianing Song, Aug 10 2022: (Start)
a(n) = A051775(2,n).
a(n) = 2*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 2*n.
a(n) = n/3 if n has only digits 0 or 3 in quaternary (n is in A001196). Otherwise, a(n) > n/3.
a(n) = 3*n/2 if and only if n has only digits 0 or 2 in quaternary (n is in A062880). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=3} 4^i, then a(n) = 3*n/2 if and only if 2*A + B = 3/2*(A + 3*B), or A = 7*B. If B != 0, then B is of the form (4*s+1)*4^t, but 7*B is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 2. (End)
MAPLE
a:= proc(n) option remember; `if`(n=0, 0,
a(iquo(n, 4, 'r'))*4+[0, 2, 3, 1][r+1])
end:
seq(a(n), n=0..70); # Alois P. Heinz, Jan 25 2022
MATHEMATICA
a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 2, 3, 1}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)
PROG
(PARI) a(n) = my(v=digits(n, 4), w=[0, 2, 3, 1]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022
(Python)
def a(n, D=[0, 2, 3, 1]):
r, k = 0, 0
while n>0: r+=D[n%4]*4**k; n//=4; k+=1
return r
# Onur Ozkan, Mar 07 2023
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Erich Friedman.
STATUS
approved