A004468 a(n) = Nim product 3 * n.

0, 3, 1, 2, 12, 15, 13, 14, 4, 7, 5, 6, 8, 11, 9, 10, 48, 51, 49, 50, 60, 63, 61, 62, 52, 55, 53, 54, 56, 59, 57, 58, 16, 19, 17, 18, 28, 31, 29, 30, 20, 23, 21, 22, 24, 27, 25, 26, 32, 35, 33, 34, 44, 47, 45, 46, 36, 39, 37, 38, 40, 43, 41, 42, 192, 195, 193, 194, 204, 207, 205

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• look

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)
• Index entries for sequences related to Nim-multiplication
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Row 3 of array in A051775.

Cf. A006015, A004469-A004480, A000695, A062880, A001196.

Programs

Maple

read("transforms") ;
# insert Maple procedures nimprodP2() and A051775() of the b-file in A051775 here.
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 *)

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

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)

Extensions

More terms from Erich Friedman