A126006 -id:A126006 - cp's OEIS frontend

A057300 Binary counter with odd/even bit positions swapped; base-4 counter with 1's replaced by 2's and vice versa.

0, 2, 1, 3, 8, 10, 9, 11, 4, 6, 5, 7, 12, 14, 13, 15, 32, 34, 33, 35, 40, 42, 41, 43, 36, 38, 37, 39, 44, 46, 45, 47, 16, 18, 17, 19, 24, 26, 25, 27, 20, 22, 21, 23, 28, 30, 29, 31, 48, 50, 49, 51, 56, 58, 57, 59, 52, 54, 53, 55, 60, 62, 61, 63, 128, 130, 129, 131, 136, 138

Offset: 0

Marc LeBrun, Aug 24 2000

A self-inverse permutation of the integers.
a(n) = n if and only if n can be written as 3*Sum_{k>=0} d_i*4^k, where d_i is either 0 or 1. - Jon Perry, Oct 06 2012
From Veselin Jungic, Mar 03 2015: (Start)
In 1988 A. F. Sidorenko, see the Sidorenko reference, used this sequence as an example of a permutation of the set of positive integers with the property that if positive integers i, j, and k form a 3-term arithmetic progression then the corresponding terms a(i), a(j), and a(k) do not form an arithmetic progression.
In the terminology introduced in the Brown, Jungic, and Poelstra reference, the sequence does not contain "double 3-term arithmetic progressions".
It is not difficult to check that this sequence is with unbounded gaps, i.e., for any positive number m there is a natural number n such that a(n+1) - a(n) > m.
It is an open question if every sequence of integers with bounded gaps must contain a double 3-term arithmetic progression. This problem is equivalent to the well known additive square problem in infinite words: Is it true that any infinite word with a finite set of integers as its alphabet contains two consecutive blocks of the same length and the same sum? For more details about the additive square problem in infinite words see the following references: Ardal, et al.; Brown and Freedman; Freedman; Grytczuk; Halbeisen and Hungerbuhler, and Pirillo and Varricchio.
The sequence was attributed to Sidorenko in P. Hegarty's paper "Permutations avoiding arithmetic patterns". In his paper Hegarty characterized the countably infinite abelian groups for which there exists a bijection mapping arithmetic progressions to non-arithmetic progressions. This was further generalized by Jungic and Sahasrabudhe. (End)

			a(31) = a(4*7+3) = 4*a(7) + a(3) = 4*11 + 3 = 47.

Paul Tek, Table of n, a(n) for n = 0..16383
H. Ardal, T. Brown, V. Jungic, and J. Sahasrabudhe, On Additive and Abelian Complexity in Infinite Words, INTEGERS: Elect. J. Combin. Number Theory, 12 (2012), A21.
T. C. Brown and A. R. Freedman, Arithmetic progressions in lacunary sets, Rocky Mountain J. Math., 17 Number 3 (1987), 587-596. doi:10.1216/RMJ-1987-17-3-587
T. Brown, V. Jungic, and A. Poelstra, On 3-term double arithmetic progressions, INTEGERS: Elect. J. Combin. Number Theory, 14 (2014), A43.
A. R. Freedman, Sequences on sets of four numbers, INTEGERS: Elect. J. Combin. Number Theory, 16 (2016), A33.
J. Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
L. Halbeisen and N. Hungerbuhler, An application of van der Waerden's theorem in additive number theory, INTEGERS: Elect. J. Combin. Number Theory, 0 (2000), A7.
Peter Hegarty, Permutations avoiding arithmetic patterns, The Electronic Journal of Combinatorics, 11 (2004), #R39.
V. Jungic and J. Sahasrabudhe, Permutations destroying arithmetic structure, The Electronic Journal of Combinatorics, Volume 22, Issue 2 (2015), Paper #P2.5.
G. Pirillo and S. Varricchio, On uniformly repetitive semigroups, Semigroup Forum, 49 (1994), 125-129.
A. F. Sidorenko, An infinite permutation without arithmetic progressions, Discrete Math., 69 (1988), 211.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Sequences used in definitions of this sequence: A000695, A059905, A059906.
Sequences with similar definitions: A057301, A126006, A126007, A126008, A163241, A163327.
A003986, A003987, A004198, A053985, A054240 are used to express relationships between sequence terms.

#include 
uint32_t a(uint32_t n) { return ((n & 0x55555555) << 1) | ((n & 0xaaaaaaaa) >> 1); } /* Falk Hüffner, Jan 23 2022 */

a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 2, 1, 3][r+1])
    end:
seq(a(n), n=0..69);  # Alois P. Heinz, Jan 25 2022

Table[FromDigits[IntegerDigits[n,4]/.{1->2,2->1},4],{n,0,70}] (* Harvey P. Dale, Aug 24 2017 *)

A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); }; \\ Antti Karttunen, Apr 14 2018

Conjecture: a(2*n) = -2*a(n) + 5*n, a(2*n+1) = -2*a(n) + 5*n + 2. - Ralf Stephan, Oct 11 2003
a(4n+k) = 4a(n) + a(k), 0 <= k <= 3. - Jon Perry, Oct 06 2012
a(n) = A000695(A059906(n)) + 2*A000695(A059905(n)). - Antti Karttunen, Apr 14 2018
From Peter Munn, Dec 10 2019: (Start)
a(a(n)) = n.
a(A000695(m) + 2*A000695(n)) = 2*A000695(m) + A000695(n).
a(n OR k) = a(n) OR a(k), where OR is bitwise-or (A003986).
a(n XOR k) = a(n) XOR a(k), where XOR is bitwise exclusive-or (A003987).
a(n AND k) = a(n) AND a(k), where AND is bitwise-and (A004198).
a(A054240(n,k)) = A054240(a(n), a(k)). (End)
a(n) = 5*n/4 - 3*A053985(2*n)/8. - Alan Michael Gómez Calderón, May 20 2025

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '(000...)102' (... + 0*27 + 1*9 + 0*3 + 2), which results '1020' = 1*27 + 0*9 + 2*3 + 0 = 33, when the odd- and even-positioned digits are swapped, thus a(11) = 33.

Antti Karttunen, Table of n, a(n) for n = 0..728
Index entries for sequences that are permutations of the natural numbers

Cf. A007089, A163328-A163329.

Other bases: A057300, A126006, A217558.

from sympy.ntheory import digits
def a(n):
    d = digits(n, 3)[1:]
    return sum(3**(i+(1-2*(i&1)))*di for i, di in enumerate(d[::-1]))
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 05 2022

(define (A163327 n) (+ (A037314 (A163326 n)) (* 3 (A037314 (A163325 n)))))

a(n) = A037314(A163326(n)) + 3*A037314(A163325(n))

Edited by Charles R Greathouse IV, Nov 01 2009

A126007 Involution of nonnegative integers: Keep the least significant quaternary digit q0 of n fixed, but swap the positions of digits q1 <-> q2, q3 <-> q4, ..., etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).

Original entry on oeis.org

0, 1, 2, 3, 16, 17, 18, 19, 32, 33, 34, 35, 48, 49, 50, 51, 4, 5, 6, 7, 20, 21, 22, 23, 36, 37, 38, 39, 52, 53, 54, 55, 8, 9, 10, 11, 24, 25, 26, 27, 40, 41, 42, 43, 56, 57, 58, 59, 12, 13, 14, 15, 28, 29, 30, 31, 44, 45, 46, 47, 60, 61, 62, 63, 256, 257, 258, 259, 272

Offset: 0

Antti Karttunen, Jan 02 2007

Index entries for sequences that are permutations of the natural numbers

Cf. A057300, A126006, A126008.

#include 
uint32_t a(uint32_t n) { return ((n & 0xcccccccc) << 2) | ((n & 0x33333330) >> 2) | (n & 3); } /* Falk Hüffner, Jan 23 2022 */

f(n) = my(d=Vecrev(digits(n, 4))); if (#d % 2, d = concat(d, 0)); fromdigits(Vecrev(vector(#d, i, d[i+(-1)^(i-1)])), 4); \\ A126006
a(n) = (n % 4) + 4*f(n\4); \\ Michel Marcus, Jan 23 2022

a(n) = (n mod 4) + 4*A126006(floor(n/4)).

a(n) = A057300(A126008(n)) = A126008(A057300(n)).

A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 3, 3, 4, 0, 0, 5, 8, 0, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 9, 15, 1, 15, 9, 7, 0, 0, 8, 11, 15, 5, 5, 15, 11, 8, 0, 0, 9, 4, 12, 9, 0, 9, 12, 4, 9, 0, 0, 10, 6, 12, 13, 15, 15, 13, 12, 6, 10, 0

Offset: 0

Rémy Sigrist, Feb 08 2021

For any x >= 0, the function n -> T(n, 2^x) is a self-inverse permutation of the nonnegative integers.
The set of nonnegative integers equipped with T forms a commutative monoid; its invertible elements are the odious numbers (A000069).
Hence A000069 equipped with T forms a group.

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+---------------------------------------------------------------
    0|  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
    1|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    2|  0   2   1   3   8  10   9  11   4   6   5   7  12  14  13  15 -> A057300
    3|  0   3   3   0  12  15  15  12  12  15  15  12   0   3   3   0
    4|  0   4   8  12   1   5   9  13   2   6  10  14   3   7  11  15 -> A126006
    5|  0   5  10  15   5   0  15  10  10  15   0   5  15  10   5   0
    6|  0   6   9  15   9  15   0   6   6   0  15   9  15   9   6   0
    7|  0   7  11  12  13  10   6   1  14   9   5   2   3   4   8  15
    8|  0   8   4  12   2  10   6  14   1   9   5  13   3  11   7  15
    9|  0   9   6  15   6  15   0   9   9   0  15   6  15   6   9   0
   10|  0  10   5  15  10   0  15   5   5  15   0  10  15   5  10   0
   11|  0  11   7  12  14   5   9   2  13   6  10   1   3   8   4  15
   12|  0  12  12   0   3  15  15   3   3  15  15   3   0  12  12   0
   13|  0  13  14   3   7  10   9   4  11   6   5   8  12   1   2  15
   14|  0  14  13   3  11   5   6   8   7   9  10   4  12   2   1  15
   15|  0  15  15   0  15   0   0  15  15   0   0  15   0  15  15   0
                                                                     \
                                                                      v
                                                                    A010060

Rémy Sigrist, Table of n, a(n) for n = 0..10010 (rows 0..140)
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Rémy Sigrist, Colored representation of the table over the first 128 odious numbers
Rémy Sigrist, Colored representation of the table over the first 1024 evil numbers (white pixels correspond to 0's)

Cf. A000069, A010060, A057300, A126006, A133457.

B(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n -= 2^(b[k] = valuation(n, 2))); b }
T(n,k) = { my (nn=B(n), kk=B(k), v=0); for (i=1, #nn, for (j=1, #kk, v=bitxor(v, 2^bitxor(nn[i], kk[j])))); v }

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, 0) = 0 (0 is an absorbing element for T).
T(n, 1) = n (1 is the neutral element for T).
T(n, 2) = A057300(n).
T(n, 4) = A126006(n).
T(n, n) = A010060(n).
A010060(T(n, k)) = A010060(n) * A010060(k).

cp's OEIS Frontend

A057300 Binary counter with odd/even bit positions swapped; base-4 counter with 1's replaced by 2's and vice versa.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Maple

Mathematica

PARI

Formula

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Scheme

Formula

Extensions

A126007 Involution of nonnegative integers: Keep the least significant quaternary digit q0 of n fixed, but swap the positions of digits q1 <-> q2, q3 <-> q4, ..., etc. in the base-4 expansion of n (where n = ... + q4*256 + q3*64 + q2*16 + q1*4 + q0).

Views

Author

Keywords

Links

Crossrefs

Programs

C

PARI

Formula

A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A341489 Third row of A341458.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A126007 Involution of nonnegative integers: Keep the least significant quaternary digit q0 of n fixed, but swap the positions of digits q1 <-> q2, q3 <-> q4, ..., etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).