OFFSET
0,2
COMMENTS
From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 3,1,2.
This is a permutation of the natural numbers; A006015 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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
FORMULA
a(n) = A051775(3,n).
From Jianing Song, Aug 10 2022: (Start)
a(n) = 3*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 3*n.
a(n) = n/2 if n has only digits 0 or 2 in quaternary (n is in A062880). Otherwise, a(n) > n/2.
a(n) = 2*n/3 if and only if n has only digits 0 or 3 in quaternary (n is in A001196). 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=2} 4^i, then a(n) = 2*n/3 if and only if 3*A + B = 2/3*(A + 2*B), or B = 7*A. If A != 0, then A is of the form (4*s+1)*4^t, but 7*A is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 3. (End)
MAPLE
read("transforms") ;
A004468 := proc(n)
A051775(3, n) ;
end proc:
L := [seq(A004468(n), n=0..1000)] ; # R. J. Mathar, May 28 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
a(iquo(n, 4, 'r'))*4+[0, 3, 1, 2][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, 3, 1, 2}[[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, 3, 1, 2]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022
(Python)
def a(n, D=[0, 3, 1, 2]):
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