A004443 Nimsum n + 2.

2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 18, 19, 16, 17, 22, 23, 20, 21, 26, 27, 24, 25, 30, 31, 28, 29, 34, 35, 32, 33, 38, 39, 36, 37, 42, 43, 40, 41, 46, 47, 44, 45, 50, 51, 48, 49, 54, 55, 52, 53, 58, 59, 56, 57, 62, 63, 60, 61, 66, 67, 64, 65

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A self-inverse permutation of the natural numbers. - Philippe Deléham, Nov 22 2016

References

• E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
• J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..20000
• F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
• Index entries for sequences related to Nim-sums
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A003987, A004442, A269526.

Essentially the same as A256008 - 1.

Also the second column of A274528.

Cf. A002162.

Programs

Maple

nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^200,base,2); t2 := convert(b+2^200,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end;
f := n -> n + 2*(-1)^floor(n/2); # N. J. A. Sloane, Jul 06 2019

Mathematica

Table[BitXor[n, 2], {n, 0, 100}] (* T. D. Noe, Feb 09 2013 *)

PARI

a(n)=bitxor(n,2) \\ Charles R Greathouse IV, Oct 07 2015

Python

for n in range(20): print(2^n) # Oliver Knill, Feb 16 2020

Formula

a(n) = n XOR 2. - Joerg Arndt, Feb 07 2013

G.f.: (2-x-2x^2+3x^3)/((1-x)^2(1+x^2)). - Ralf Stephan, Apr 24 2004

The sequences 'Nimsum n + m' seem to have the general o.g.f. p(x)/q(x) with p, q polynomials and q(x) = (1-x)^2*Product_{k>=0} (1+x^(2^e(k))), with Sum_{k>=0} 2^e(k) = m. - Ralf Stephan, Apr 24 2004

a(n) = n + 2(-1)^floor(n/2). - Mitchell Harris, Jan 10 2005

a(n) = OR(n,2) - AND(n,2). - Gary Detlefs, Feb 06 2013

E.g.f.: 2*(sin(x) + cos(x)) + x*exp(x). - Ilya Gutkovskiy, Jul 01 2016

Sum_{n>=0,n<>2} (-1)^n/a(n) = -log(2) = -A002162. - Peter McNair, Aug 07 2023