A080412 Exchange rightmost two binary digits of n > 1; a(0)=0, a(1)=2.

0, 2, 1, 3, 4, 6, 5, 7, 8, 10, 9, 11, 12, 14, 13, 15, 16, 18, 17, 19, 20, 22, 21, 23, 24, 26, 25, 27, 28, 30, 29, 31, 32, 34, 33, 35, 36, 38, 37, 39, 40, 42, 41, 43, 44, 46, 45, 47, 48, 50, 49, 51, 52, 54, 53, 55, 56, 58, 57, 59, 60, 62, 61, 63, 64, 66, 65, 67, 68, 70, 69, 71, 72

Offset: 0

Author

Reinhard Zumkeller, Feb 17 2003

Keywords

• nonn
• easy
• nice

Comments

Self-inverse permutation of the natural numbers: a(a(n)) = n.

Lodumo_2 of A021913. - Philippe Deléham, Apr 26 2009

The lodumo_m transformation of a list L is the list L' such that L'(n) is the smallest nonnegative integer not occurring earlier in L' and equal to L(n) (mod m). - M. F. Hasler, Dec 06 2010

From Franck Maminirina Ramaharo, Jul 20 2018: (Start)

Let

A: 0, 3, 8, 11, 16, 19, 24, 27, 32, 35, 40, 43, 48, 51, 56, 59, ... A047470

B: 1, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, 46, 49, 54, 57, 62, ... A047452

C: 2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, 45, 50, 53, 58, 61, ... A047617

D: 4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, 47, 52, 55, 60, 63, ... A047535.

Then the sequence is obtained by repeatedly picking terms from A,B,C,D according to the circuit A-C-B-A-D-B-C-D. The sequence begins:

A | C | B | A | D | B | C | D || A | C | B | A | D | ...

--+---+---+---+---+---+---+---++---+---+---+---+---+----

0 | 2 | 1 | 3 | 4 | 6 | 5 | 7 || 8 |10 | 9 |11 |12 | ...

(End)

The sequence is a permutation of the nonnegative integers partitioned into quadruples [4k, 4k+2, 4k+1, 4k+3] for k >= 0, i.e., the two interior terms of each quadruple are interchanged. - Guenther Schrack, Apr 22 2019

Examples

			a(20) = a('101'00') = '101'00' = 20; a(21) = a('101'01') = '101'10' = 22.
a(2) = a('10') = '01' = 1; a(3) = a('11') = '11' = 3.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A004442, A007088, A021913, A080413, A080414.

Cf. A047470, A047452, A047617, A047535.

Programs

GAP

a:=[0,2,1,3,4];; for n in [6..80] do a[n]:=a[n-1]+a[n-4]-a[n-5]; od; a; # Muniru A Asiru, Jul 27 2018

Magma

R:=PowerSeriesRing(Integers(), 80); [0] cat Coefficients(R!( x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4)) )); // G. C. Greubel, Apr 28 2019

Maple

A080412:=n->n+1+(1+I)*(2*I-2-(1-I)*I^(2*n)+I^(-n)-I^(1+n))/4: seq(A080412(n), n=0..100); # Wesley Ivan Hurt, May 28 2016

Mathematica

a[n_] := (bits = IntegerDigits[n, 2]; Join[Drop[bits, -2], {bits[[-1]], bits[[-2]]}] // FromDigits[#, 2]&); a[0]=0; a[1]=2; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Mar 11 2013 *)
ertbd[n_]:=Module[{a,b},{a,b}=TakeDrop[IntegerDigits[n,2], IntegerLength[ n,2]-2];FromDigits[Join[a,Reverse[b]],2]]; Join[{0,2},Array[ertbd,80,2]] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jan 07 2016 *)
CoefficientList[Series[x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4)), {x,0,80}], x] (* G. C. Greubel, Apr 28 2019 *)

PARI

my(x='x+O('x^80)); concat([0], Vec(x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Apr 28 2019

Python

def A080412(n): return (0,1,-1,0)[n&3]+n # Chai Wah Wu, Jan 18 2023

Sage

(x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4))).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

a(n) = 4*floor(n/4) + a(n mod 4), for n > 3.

a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4. - Joerg Arndt, Mar 11 2013

a(n) = lod_2(A021913(n)). - Philippe Deléham, Apr 26 2009

From Wesley Ivan Hurt, May 28 2016: (Start)

a(n) = n + 1 + (1+i)*(2*i-2-(1-i)*i^(2*n) + i^(-n)-i^(1+n))/4 where i=sqrt(-1).

G.f.: x*(2-x+2*x^2+x^3) / ((1-x)^2*(1+x+x^2+x^3)). (End)

E.g.f.: (sin(x) + cos(x) + (2*x + 1)*sinh(x) + (2*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 28 2016

From Guenther Schrack, Apr 23 2019: (Start)

a(n) = (2*n - (-1)^n + (-1)^(n*(n-1)/2))/2.

a(n) = a(n-4) + 4, a(0)=0, a(1)=2, a(2)=1, a(3)=3, for n > 3. (End)

Extensions

Typo in example fixed by Reinhard Zumkeller, Jul 06 2009