A021913 -id:A021913 - cp's OEIS frontend

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

Reinhard Zumkeller, Feb 17 2003

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

			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.

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

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

Cf. A047470, A047452, A047617, A047535.

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

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

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

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 *)

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

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

(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

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)

Typo in example fixed by Reinhard Zumkeller, Jul 06 2009

A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

Original entry on oeis.org

1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1

Offset: 0

Paul Barry, May 15 2003

Partial sums of A084099. Inverse binomial transform of A000749 (without leading zeros).
From Klaus Brockhaus, May 31 2010: (Start)
Periodic sequence: Repeat 1, 3, 3, 1.
Interleaving of A010684 and A176040.
Continued fraction expansion of (7 + 5*sqrt(29))/26.
Decimal expansion of 121/909.
a(n) = A143432(n+3) + 1 = 2*A021913(n+1) + 1 = 2*A133872(n+3) + 1.
a(n) = A165207(n+1) - 1.
First differences of A047538.
Binomial transform of A084102. (End)
From Wolfdieter Lang, Feb 09 2012: (Start)
a(n) = A045572(n+1) (Modd 5) := A203571(A045572(n+1)), n >= 0.
For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the five residue classes Modd 5, called [m] for m=0,1,...,4, are shown in the array A090298 if there the last row is taken as class [0] after inclusion of 0.
(End)

			From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 5 of nonnegative odd numbers restricted mod 5:
A045572: 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, ...
Modd 5:  1, 3, 3, 1,  1,  3,  3,  1,  1,  3, ...
(End)

Guenther Schrack, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1).

Cf. A084102.

Cf. A010684 (repeat 1, 3), A176040 (repeat 3, 1), A178593 (decimal expansion of (7+5*sqrt(29))/26), A143432 (expansion of (1+x^4)/((1-x)*(1+x^2))), A021913 (repeat 0, 0, 1, 1), A133872 (repeat 1, 1, 0, 0), A165207 (repeat 2, 2, 4, 4), A047538 (congruent to 0, 1, 4 or 7 mod 8), A084099 (expansion of (1+x)^2/(1+x^2)), A000749 (expansion of x^3/((1-x)^4-x^4)). - Klaus Brockhaus, May 31 2010

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x)^2/((1-x)*(1+x^2)) )); // G. C. Greubel, Feb 28 2019

CoefficientList[Series[(1+x)^2/((1-x)(1+x^2)),{x,0,110}],x] (* or *) PadRight[{},110,{1,3,3,1}] (* Harvey P. Dale, Nov 21 2012 *)

x='x+O('x^100); Vec((1+x)^2/((1-x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

((1+x)^2/((1-x)*(1+x^2))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 28 2019

a(n) = binomial(3, n mod 4). - Paul Barry, May 25 2003
From Klaus Brockhaus, May 31 2010: (Start)
a(n) = a(n-4) for n > 3; a(0) = a(3) = 1, a(1) = a(2) = 3.
a(n) = (4 - (1+i)*i^n - (1-i)*(-i)^n)/2 where i = sqrt(-1). (End)
E.g.f.: 2*exp(x) + sin(x) - cos(x). - Arkadiusz Wesolowski, Nov 04 2017
a(n) = 2 - (-1)^(n*(n+1)/2). - Guenther Schrack, Feb 26 2019

A145768 a(n) = the bitwise XOR of squares of first n natural numbers.

Original entry on oeis.org

0, 1, 5, 12, 28, 5, 33, 16, 80, 1, 101, 28, 140, 37, 225, 0, 256, 33, 357, 12, 412, 37, 449, 976, 400, 993, 325, 924, 140, 965, 65, 896, 1920, 961, 1861, 908, 1692, 965, 1633, 912, 1488, 833, 1445, 668, 1292, 741, 2721, 512, 2816, 609, 2981, 396, 2844, 485

Offset: 0

Vladimir Reshetnikov, Oct 18 2008

Up to n=10^8, a(15) is the only zero term and a(1)=a(9) are the only terms for which a(n)=1. Can it be proved that any number can only appear a finite number of times in this sequence? [M. F. Hasler, Oct 20 2008]
Even terms occur at A014601, odd terms at A042963; A010873(a(n))=A021913(n+1). - Reinhard Zumkeller, Jun 05 2012
If squares occur, they must be at indexes != 2 or 5 (mod 8). - Roderick MacPhee, Jul 17 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
StackExchange, Perfect squares in a XOR-Sum of perfect squares

Cf. A003815, A145827, A145828, A145829, A145830, A145831. [M. F. Hasler, Oct 20 2008]
Cf. A193232.
Cf. A000290.

import Data.Bits (xor)
a145768 n = a145768_list !! n
a145768_list = scanl1 xor a000290_list  -- Reinhard Zumkeller, Jun 05 2012

A[0]:= 0:
for n from 1 to 100 do A[n]:= Bits:-Xor(A[n-1],n^2) od:
seq(A[i],i=0..100); # Robert Israel, Dec 08 2019

Rest@ FoldList[BitXor, 0, Array[#^2 &, 50]]

an=0; for( i=1,50, print1(an=bitxor(an,i^2),",")) \\ M. F. Hasler, Oct 20 2008

al(n)=local(m);vector(n,k,m=bitxor(m,k^2))

from functools import reduce
from operator import xor
def A145768(n):
    return reduce(xor, [x**2 for x in range(n+1)]) # Chai Wah Wu, Aug 08 2014

a(n)=1^2 xor 2^2 xor ... xor n^2.

A169671 Lexicographically earliest de Bruijn sequence for n = 6 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 64, the period being:
0000001000011000101000111001001011001101001111010101110110111111.

Ray Chandler, Table of n, a(n) for n = 0..1023
Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1).

Cf. A021913, A169675, A080679, A169672, A169671, A169673, A169674.

See A058342 for another version.

A255126 Number of times a number of the form 4n+2 is encountered when iterating from 2^(n+1)-2 to (2^n)-2 with the map x -> x - (number of runs in binary representation of x).

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 10, 16, 27, 50, 97, 188, 355, 652, 1177, 2126, 3886, 7204, 13501, 25465, 48192, 91411, 173851, 331821, 636035, 1224505, 2366662, 4588124, 8913418, 17338878, 33756650, 65766474, 128239805, 250346859, 489422205, 958304970, 1879145187, 3689012737

Offset: 0

Antti Karttunen, Feb 18 2015

nonn

Also the number of odd numbers in range [A255062(n) .. A255061(n+1)] of A255057 (equally, in A255067). See the sum-formulas.

			For n=5 we start iterating with map m(n) = A236840(n) from the initial value (2^(5+1))-2 = 62. Thus we get m(62) = 60, m(60) = 58, m(58) = 54, m(54) = 50, m(50) = 46, m(46) = 42, m(42) = 36, m(36) = 32 and finally m(32) = 30, which is (2^5)-2. Of the nine numbers encountered, only 58, 54, 50, 46, 42 and 30 are of the form 4n+2, thus a(5) = 6. Note that the initial value 2^(n+1)-2 is not included in the cases, but the final (2^n) - 2 is.

Cf. A000035, A005811, A021913, A236840, A255057, A255061, A255062, A255067, A255071, A255125.

Similar sequences: A218543, A255064.

PARI
```
\\ Use the PARI-code given in A255125.
```

(define (A255126 n) (if (zero? n) n (let loop ((i (- (expt 2 (+ 1 n)) 4)) (s 1)) (cond ((pow2? (+ 2 i)) s) (else (loop (- i (A005811 i)) (+ s (A021913 i))))))))
;; Alternatively:
(define (A255126 n) (add (COMPOSE A000035 A255057) (A255062 n) (A255061 (+ 1 n))))
(define (A255126 n) (add (COMPOSE A000035 A255067) (A255062 n) (A255061 (+ 1 n))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A000035(A255057(k)).
a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A000035(A255067(k)).
a(n) = A255071(n) - A255125(n).

A083219 a(n) = n - 2*floor(n/4).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38

Offset: 0

Reinhard Zumkeller, Apr 22 2003

Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the left half-plane. - Michel Lagneau, Apr 09 2013

a(n) is n+2 with its second least significant bit removed (see A021913(n+2) for that bit). - Kevin Ryde, Dec 13 2019

Altug Alkan, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A083220, A129756, A162751 (second highest bit removed).
Cf. A021913, A162330, A285869.
Essentially the same as A018837.

a:= n-> n - 2*floor(n/4):
seq(a(n), n=0..74);  # Alois P. Heinz, Jan 24 2021

Array[# - 2 Floor[#/4] &, 75, 0] (* Michael De Vlieger, Oct 02 2017 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,2},80] (* Harvey P. Dale, Oct 01 2018 *)

a(n) = n - n\4*2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A083220(n)/2.
a(n) = a(n-1) + n mod 2 + (n mod 4 - 1)*(1 - n mod 2), a(0) = 0.
G.f.: x*(1+x+x^2-x^3)/((1-x)^2*(1+x)*(1+x^2)). - R. J. Mathar, Aug 28 2008
a(n) = n - A129756(n). - Michel Lagneau, Apr 09 2013
Bisection: a(2*k) = 2*floor((n+2)/4), a(2*k+1) = a(2*k) + 1, k >= 0. - Wolfdieter Lang, May 08 2017
a(n) = (2*n + 3 - 2*cos(n*Pi/2) - cos(n*Pi) - 2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
a(n) = A162330(n+2) - 1 = A285869(n+3) - 1. - Kevin Ryde, Dec 13 2019
E.g.f.: ((1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x))/2. - Stefano Spezia, May 27 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 1. - Amiram Eldar, Aug 21 2023

A169673 Lexicographically earliest de Bruijn sequence for n = 7 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 128, the period being:
00000001000001100001010000111000100100010110001101000111100100110\
010101001011100110110011101001111101010110101111011011101111111

Ray Chandler, Table of n, a(n) for n = 0..1023
Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, order 128.

Cf. A021913, A169675, A080679, A169672, A169671, A169674.

A169674 Lexicographically earliest de Bruijn sequence for n = 8 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 256, the period being:
0000000010000001100000101000001110000100100001011000011010000111100010\
0010011000101010001011100011001000110110001110100011111001001010010011\
1001010110010110100101111001100110101001101110011101100111101001111110\
1010101110101101101011111011011110111011111111

Ray Chandler, Table of n, a(n) for n = 0..1023
Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, order 256.

Cf. A021913, A169675, A080679, A169672, A169671, A169673.

A169672 Lexicographically earliest de Bruijn sequence for n = 5 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 32, the period being: 00000100011001010011101011011111.

Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A021913, A169675, A080679, A169671, A169673, A169674.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1},99] (* Ray Chandler, Aug 26 2015 *)

A169675 Lexicographically earliest de Bruijn sequence for n = 3 and k = 2.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

The lexicographically earliest de Bruijn sequence for n = 2 and k = 2 is 0011 repeated (see A021913).

			Periodic with period 8, the period being 00010111.

Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A021913, A080679, A169672, A169671, A169673, A169674.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{0, 0, 0, 1, 0, 1, 1, 1},99] (* Ray Chandler, Aug 25 2015 *)
PadRight[{},120,{0,0,0,1,0,1,1,1}] (* Harvey P. Dale, Aug 01 2024 *)

cp's OEIS Frontend

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

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A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

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A145768 a(n) = the bitwise XOR of squares of first n natural numbers.

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PARI

Python

Formula

A169671 Lexicographically earliest de Bruijn sequence for n = 6 and k = 2.

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A255126 Number of times a number of the form 4n+2 is encountered when iterating from 2^(n+1)-2 to (2^n)-2 with the map x -> x - (number of runs in binary representation of x).

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Programs

PARI

Scheme

Formula

A083219 a(n) = n - 2*floor(n/4).

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Maple

Mathematica

PARI

Formula

A169673 Lexicographically earliest de Bruijn sequence for n = 7 and k = 2.

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