A232751 -id:A232751 - cp's OEIS frontend

A193231 Blue code for n: in binary coding of a polynomial over GF(2), substitute x+1 for x (see Comments for precise definition).

0, 1, 3, 2, 5, 4, 6, 7, 15, 14, 12, 13, 10, 11, 9, 8, 17, 16, 18, 19, 20, 21, 23, 22, 30, 31, 29, 28, 27, 26, 24, 25, 51, 50, 48, 49, 54, 55, 53, 52, 60, 61, 63, 62, 57, 56, 58, 59, 34, 35, 33, 32, 39, 38, 36, 37, 45, 44, 46, 47, 40, 41, 43, 42, 85, 84, 86

Offset: 0

Franklin T. Adams-Watters, Jul 18 2011

This is a self-inverse permutation of the nonnegative integers.
The function "substitute x+1 for x" on polynomials over GF(2) is completely multiplicative.
What is the density of fixed points in this sequence? Do we get a different answer if we look only at irreducible polynomials?
From Antti Karttunen, Dec 27 2013: (Start)
As what comes to the above question, the number of fixed points in range [2^(n-1),(2^n)-1] of the sequence is given by A131575(n). In range [0,0] there is one fixed point: 0, in range [1,1] there is also one: 1, in range [2,3] there are no fixed points, in range [4,7] there are two fixed points: 6 and 7, and so on. (Cf. also the C-code given in A118666.)
Similarly, the number of cycles in such ranges begins as 1, 1, 1, 3, 4, 10, 16, 36, 64, 136, ... which is A051437 shifted two steps right (prepended with 1's): Because the sequence is a self-inverse permutation, the number of its cycles in range [2^(n-1),(2^n)-1] is computed as: cycles(n) = (A011782(n)-number_of_fixedpoints(n))/2 + number_of_fixedpoints(n), which matches with the identity: A051437(n-2) = (A011782(n)-A131575(n))/2 + A131575(n), for n>=2.
In OEIS terms, the above comment about multiplicativeness can be rephrased as: a(A048720(x,y)) = A048720(a(x),a(y)) for all integers x, y >= 0. Here A048720(x,y) gives the product of carryless binary multiplication of x and y.
The permutation conjugates between Gray code and its inverse: A003188(n) = a(A006068(a(n))) and A006068(n) = a(A003188(a(n))) [cf. the identity 1.19-9d: gB = Bg^{-1} given on page 53 of fxtbook].
Because of the multiplicativity, the subset of irreducible (and respectively: composite) polynomials over GF(2) is closed under this permutation. Cf. the following mappings: a(A014580(n)) = A234750(n) and a(A091242(n)) = A234745(n).
(End)

			11, binary 1011, corresponds to polynomial x^3+x+1, substituting: (x+1)^3+(x+1)+1 = x^3+x^2+x+1 + x+1 + 1 = x^3+x^2+1, binary 1101 = decimal 13, so a(11) = 13.
From _Tilman Piesk_, Jun 26 2025: (Start)
The binary exponents of 11 are {0, 1, 3}, because 11 = 2^0 + 2^1 + 2^3.
a(11) = A001317(0) XOR A001317(1) XOR A001317(3) = 1 XOR 3 XOR 15 = 13. (End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Joerg Arndt, Matters Computational (The Fxtbook), section 1.19, "Invertible transforms on words", pp. 49--55 [The name "blue code" is introduced in this book. - _Antti Karttunen_, Dec 28 2013]
Tilman Piesk, illustration of the first 256 terms
Index entries for sequences operating on (or containing) GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Cf. A000069, A001969, A001317, A003987, A048720, A048724, A065621, A051437, A118666 (fixed points), A131575, A234022 (the number of 1-bits), A234023, A010060, A234745, A234750.
Similarly constructed permutation pairs: A003188/A006068, A135141/A227413, A232751/A232752, A233275/A233276, A233277/A233278, A233279/A233280.
Other permutations based on this (by conjugating, composing, etc): A234024, A234025/A234026, A234027, A234612, A234613, A234747, A234748, A244987, A245812, A245454.

f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; Table[f@ n, {n, 0, 66}] (* Michael De Vlieger, Feb 12 2016, after Robert G. Wilson v at A048724 and A065621 *)

tox(n) = local(x=Mod(1,2)*X, xp=1, r); while(n>0,if(n%2,r+=xp);xp*=x;n\=2);r
a(n)=subst(lift(subst(tox(n),X,X+1)),X,2)

a(n)={local(x='x);subst(lift(Mod(1,2)*subst(Pol(binary(n),x),x,1+x)),x,2)};

def a065621(n): return n^(2*(n - (n&-n)))
def a048724(n): return n^(2*n)
l=[0, 1]
for n in range(2, 101):
    if n%2==0: l.append(a048724(l[n//2]))
    else: l.append(a065621(1 + l[(n - 1)//2]))
print(l) # Indranil Ghosh, Jun 04 2017

;; with memoizing macro definec available in Antti Karttunen's IntSeq-library:
(define (A193231 n) (let loop ((n n) (i 0) (s 0)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ 1 i) s)) (else (loop (/ (- n 1) 2) (+ 1 i) (A003987bi s (A001317 i))))))) ;; A003987bi implements binary XOR, A003987.
;; Antti Karttunen, Dec 27 2013

;; With memoizing macro definec available in Antti Karttunen's IntSeq-library.
;; Alternative implementation, a recurrence based on entangling even & odd numbers with complementary pair A048724 and A065621:
(definec (A193231 n) (cond ((< n 2) n) ((even? n) (A048724 (A193231 (/ n 2)))) (else (A065621 (+ (A193231 (/ (- n 1) 2)) 1)))))
;; Antti Karttunen, Dec 27 2013

From Antti Karttunen, Dec 27 2013: (Start)
a(0) = 0, and for any n = 2^a + 2^b + ... + 2^c, a(n) = A001317(a) XOR A001317(b) XOR ... XOR A001317(c), where XOR is bitwise XOR (A003987) and all the exponents a, b, ..., c are distinct, that is, they are the indices of 1-bits in the binary representation of n.
From above it follows, because all terms of A001317 are odd, that A000035(a(n)) = A010060(n) = A000035(a(2n)). Conversely, we also have A010060(a(n)) = A000035(n). Thus the permutation maps any even number to some evil number, A001969 (and vice versa), like it maps any odd number to some odious number, A000069 (and vice versa).
a(0)=0, a(1)=1, and for n>1, a(2n) = A048724(a(n)), a(2n+1) = A065621(1+a(n)). [A recurrence based on entangling even & odd numbers with the complementary pair A048724/A065621]
For all n, abs(a(2n)-a(2n+1)) = 1.
a(A000079(n)) = A001317(n).
(End)
It follows from the first paragraph above that a(A003987(n,k)) = A003987(a(n), a(k)), that is a(n XOR k) = a(n) XOR a(k). - Peter Munn, Nov 27 2019

A005228 Sequence and first differences (A030124) together list all positive numbers exactly once.

Original entry on oeis.org

1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, 285, 312, 340, 369, 399, 430, 462, 495, 529, 565, 602, 640, 679, 719, 760, 802, 845, 889, 935, 982, 1030, 1079, 1129, 1180, 1232, 1285, 1339, 1394, 1451, 1509, 1568, 1628, 1689

Offset: 1

N. J. A. Sloane

This is the lexicographically earliest sequence that together with its first differences (A030124) contains every positive integer exactly once.
Hofstadter introduces this sequence in his discussion of Scott Kim's "FIGURE-FIGURE" drawing. - N. J. A. Sloane, May 25 2013
A225850(a(n)) = 2*n-1, cf. A167151. - Reinhard Zumkeller, May 17 2013
In view of the definition of A075326: start with a(0) = 0, and extend by rule that the next term is the sum of the predecessor and the most recent non-member of the sequence. - Reinhard Zumkeller, Oct 26 2014

			Sequence reads 1 3 7 12 18 26 35 45..., differences are 2 4 5, 6, 8, 9, 10 ... and the point is that every number not in the sequence itself appears among the differences. This property (together with the fact that both the sequence and the sequence of first differences are increasing) defines the sequence!

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 73.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10001 [The first 1000 terms were computed by T. D. Noe]
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Catalin Francu, C++ program
Cristian Francu, C program to generate the N-th element in O(sqrt(N))
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
Benoit Jubin, Asymptotic series for Hofstadter's figure-figure sequences, arXiv:1404.1791; J. Integer Sequences, 17 (2014), #14.7.2.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Eric Weisstein's World of Mathematics, Hofstadter Figure-Figure Sequence.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for Hofstadter-type sequences

Cf. A030124 (complement), A037257, A056731, A056738, A140778, A225687.
The following are a group of related sequences: A005132, A006509, A037257, A037258, A037259, A081145, A093903, A099004, A100707, A129198, A129199, A140778, A225376, A225377, A225378, A225385, A225386, A225387.
Cf. A075326, A095115.
Cf. A225850, A232746, A232747 (inverse), A232739, A232740, A232750 and also permutation pair A232751/A232752 constructed from this sequence and its complement.
Cf. A001651 (analog with sums instead of differences), A121229 (analog with products).
The same recurrence a(n) = a(n-1) + c(n-1) with different starting conditions: A061577 (starting with 2), A022935 (3), A022936 (4), A022937 (5), A022938 (6).
Related recurrences:
a(n-1) + c(n+1) - A022953, A022954.
a(n-1) + c(n) - A022946 to A022952.
a(n-1) + c(n-2) - A022940, A022941.
a(n-2) + c(n-1) - A022942 to A022944.
a(n-2) + c(n-2) - A022939.
a(n-3) + c(n-3) - A022955.
a(n-4) + c(n-4) - A022956.
a(n-5) + c(n-5) - A022957.

a005228 = scanl (+) 1 a030124
a030124 = go 1 a005228 where go x ys | x < head ys = x     : go (x + 1) ys
                                     | otherwise   = x + 1 : go (x + 2) (tail ys)
-- Maks Verver, Jun 30 2025

maxn := 5000; h := array(1..5000); h[1] := 1; a := [1]; i := 1; b := []; for n from 2 to 1000 do if h[n] <> 1 then b := [op(b), n]; j := a[i]+n; if j < maxn then a := [op(a),j]; h[j] := 1; i := i+1; fi; fi; od: a; b; # a is A005228, b is A030124.
A030124 := proc(n)
    option remember;
    local a,fnd,t ;
    if n <= 1 then
        op(n+1,[2,4]) ;
    else
        for a from procname(n-1)+1 do
            fnd := false;
            for t from 1 to n+1 do
                if A005228(t)  = a then
                    fnd := true;
                    break;
                end if;
            end do:
            if not fnd then
                return a;
            end if;
        end do:
    end if;
end proc:
A005228 := proc(n)
    option remember;
    if n <= 2 then
        op(n,[1,3]) ;
    else
        procname(n-1)+A030124(n-2) ;
    end if;
end proc: # R. J. Mathar, May 19 2013

a = {1}; d = 2; k = 1; Do[ While[ Position[a, d] != {}, d++ ]; k = k + d; d++; a = Append[a, k], {n, 1, 55} ]; a
(* Second program: *)
(* Program from Larry Morris, Jan 19 2017: *)
d = 3; a = {1, 3, 7, 12, 18}; While[ Length[a = Join[a, a[[-1]] + Accumulate[Range[a[[d]] + 1, a[[++d]] - 1]]]] < 50]; a
(* Comment: This adds as many terms to the sequence as there are numbers in each set of sequential differences. Consequently, the list of numbers it produces may be longer than the limit provided. With the limit of 50 shown, the sequence produced has length 60. *)

A005228(n,print_all=0,s=1,used=0)={while(n--,used += 1<M. F. Hasler, Feb 05 2013

a(n) = a(n-1) + c(n-1) for n >= 2, where a(1)=1, a( ) increasing, c( ) = complement of a( ) (c is the sequence A030124).
Let a(n) = this sequence, b(n) = A030124 prefixed by 0. Then b(n) = mex{ a(i), b(i) : 0 <= i < n}, a(n) = a(n-1) + b(n) + 1. (Fraenkel)
a(1) = 1, a(2) = 3; a( ) increasing; for n >= 3, if a(q) = a(n-1)-a(n-2)+1 for some q < n then a(n) = a(n-1) + (a(n-1)-a(n-2)+2), otherwise a(n) = a(n-1) + (a(n-1)-a(n-2)+1). - Albert Neumueller (albert.neu(AT)gmail.com), Jul 29 2006
a(n) = n^2/2 + n^(3/2)/(3*sqrt(2)) + O(n^(5/4)) [proved in Jubin link]. - Benoit Jubin, May 13 2015
For all n >= 1, A232746(a(n)) = n and A232747(a(n)) = n. [Both sequences work as left inverses of this sequence.] - Antti Karttunen, May 14 2015

Additional comments from Robert G. Wilson v, Oct 24 2001

Incorrect formula removed by Benoit Jubin, May 13 2015

A233275 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with even and odd numbers.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 14, 10, 15, 11, 9, 8, 24, 25, 26, 28, 27, 29, 20, 30, 21, 22, 18, 31, 23, 19, 17, 16, 48, 49, 50, 52, 51, 53, 56, 54, 57, 58, 40, 55, 59, 41, 60, 42, 61, 44, 36, 43, 45, 62, 46, 37, 38, 34, 63, 47, 39, 35, 33, 32, 96, 97, 98, 100

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

It seems that for all n, a(A000079(n)) = A003945(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python Program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233276.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233277/A233278, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=1, a(n) = 2*a(A213714(n)-1), else a(n) = 1+(2*a(A234017(n))).

a(n) = A054429(A233277(n)). [Follows from the definitions of these sequences]

A233276 a(0)=0, a(1)=1, after which a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 11, 13, 8, 9, 10, 12, 31, 30, 26, 29, 22, 24, 25, 28, 16, 17, 18, 20, 19, 21, 23, 27, 63, 62, 57, 61, 50, 55, 56, 60, 42, 45, 47, 51, 49, 52, 54, 59, 32, 33, 34, 36, 35, 37, 39, 43, 38, 40, 41, 44, 46, 48, 53, 58, 127, 126, 120

Offset: 0

Antti Karttunen, Dec 18 2013

For all n, a(A000079(n)) = A000225(n+1), i.e. a(2^n) = (2^(n+1))-1.
For n>=1, a(A000225(n)) = A000325(n).
This permutation is obtained by "entangling" even and odd numbers with complementary pair A005187 & A055938, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A005187(n+1) to the parent node containing n, and each child to the right is obtained as A055938(n):
                                     0
                                     |
                  ...................1...................
                 3                                       2
       7......../ \........6                   4......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  15       14         11       13          8       9          10       12
31  30   26  29     22  24   25  28      16 17   18 20      19  21   23  27
etc.
For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to 0 from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.
Permutation A233278 gives the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233275.
Cf. A005187, A054429, A055938, A256991, A256478, A256479.
Cf. also A070939 (the binary width of both n and a(n)).
Related arrays: A255555, A255557.
Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233277/A233278, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).
As a composition of related permutations:
a(n) = A233278(A054429(n)).

Name changed and the illustration of binary tree added by Antti Karttunen, Apr 19 2015

A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 9, 13, 8, 12, 14, 15, 23, 22, 21, 19, 20, 18, 27, 17, 26, 25, 29, 16, 24, 28, 30, 31, 47, 46, 45, 43, 44, 42, 39, 41, 38, 37, 55, 40, 36, 54, 35, 53, 34, 51, 59, 52, 50, 33, 49, 58, 57, 61, 32, 48, 56, 60, 62, 63, 95, 94, 93, 91

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Indranil Ghosh, PARI program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233278.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=0, a(n) = 2*a(A234017(n)), else a(n) = 1+(2*a(A213714(n)-1)).

a(n) = A054429(A233275(n)). [Follows from the definitions of these sequences]

A233278 a(0)=0, a(1)=1, after which a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 12, 10, 9, 8, 13, 11, 14, 15, 27, 23, 21, 19, 20, 18, 17, 16, 28, 25, 24, 22, 29, 26, 30, 31, 58, 53, 48, 46, 44, 41, 40, 38, 43, 39, 37, 35, 36, 34, 33, 32, 59, 54, 52, 49, 51, 47, 45, 42, 60, 56, 55, 50, 61, 57, 62, 63, 121, 113, 108

Offset: 0

Antti Karttunen, Dec 18 2013

This permutation is obtained by "entangling" even and odd numbers with complementary pair A055938 & A005187, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A055938(n) to the parent node containing n, and each child to the right is obtained as A005187(n+1):
                                     0
                                     |
                  ...................1...................
                 2                                       3
       5......../ \........4                   6......../ \........7
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  12       10          9       8          13       11         14       15
27  23   21  19      20 18   17 16      28  25   24  22     29  26   30  31
etc.
For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to zero at the root from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.
Permutation A233276 gives the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233277.
Cf. A005187, A054429, A055938, A256991, A256478, A256479.
Cf. also A070939 (the binary width of both n and a(n)).
Related arrays: A255555, A255557.
Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).
As a composition of related permutations:
a(n) = A233276(A054429(n)).

Name changed and the illustration of binary tree added by Antti Karttunen, Apr 19 2015

A232752 Permutation of natural numbers: a(0)=0, a(1)=1, a(2n)=A005228(1+(a(n))), a(2n+1)=A030124(a(n)).

Original entry on oeis.org

0, 1, 3, 2, 12, 5, 7, 4, 114, 16, 26, 8, 45, 10, 18, 6, 7562, 127, 191, 21, 462, 32, 56, 11, 1285, 53, 83, 14, 236, 23, 35, 9, 29172079, 7677, 9314, 141, 20528, 208, 312, 27, 115291, 489, 679, 39, 1943, 65, 98, 15, 865555, 1331, 1751, 62, 4111, 94, 150, 19, 30983, 255, 369, 29, 802, 42, 69, 13

Offset: 0

Antti Karttunen, Nov 30 2013

nonn

This is one example of the generic class of "entangling of two pairs of complementary sets" permutations of natural numbers. In this case, the Hofstadter's complementary pair A005228 & A030124 is entangled with complementary pair of A005843 & A005408, the even & odd numbers.
Note how, apart from 1, all the other terms of A005228 (1, 3, 7, 12, 18, 26, ...) occur in even positions, and all the terms of A030124 (2, 4, 5, 6, 8, 9, 10, 11, 13, 14, ...) occur in odd positions.
Moreover, at the positions given by two's powers, from 2^1 = 2 onwards, a(2^n) = 3, 12, 114, 7562, 29172079, ... the values are iterates of function b(n) = A005228(n+1) from b(1)=3 onward: b(1)=3, b(b(1))=12, b(b(b(1)))=114, b(b(b(b(1))))=7562, and so on.
In the same way, at the positions given by A000225, from 2^2 - 1 = 3 onwards, the iterates of A030124 appear, A030124(1), A030124(A030124(1)), A030124(A030124(A030124(1))), and so on, as: 2, 4, 6, 9, 13, 17, ... (= A232739).
The permutation A227413 is obtained in analogous way by entangling primes and composites with even and odd numbers.

Inverse permutation: A232751.
Cf. A227413, A005228, A030124, A005843 & A005408, A000079, A000225, A232739.
Cf. also the permutation pair A167151 & A225850.

a(0)=0, a(1)=1, and for even n > 1, a(n) = A005228(1+(a(n/2))), for odd n > 1, a(n) = A030124(a((n-1)/2)).

For all n >=1, a(A000225(n+1)) = A232739(n).

A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

This permutation transforms the enumeration system of positive irreducible fractions A071766/A229742 (HCS) into the enumeration system A007305/A047679 (Stern-Brocot), and the enumeration system A245325/A245326 into A162909/A162910 (Bird). - Yosu Yurramendi, Jun 09 2015

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233279.
Cf. A000069, A001969, A003188, A054429, A063946, A154436.
Similarly constructed permutation pairs: A003188/A006068, A135141/A227413, A232751/A232752, A233275/A233276, A233277/A233278, A193231 (self-inverse).

from sympy import floor
def a003188(n): return n^(n>>1)
def a054429(n): return 1 if n==1 else 2*a054429(floor(n/2)) + 1 - n%2
def a(n): return 0 if n==0 else a003188(a054429(n)) # Indranil Ghosh, Jun 11 2017

maxrow <- 8 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+    k] <- a[2^m+      k] + 2^m
a[2^(m+1)+2^m+k] <- a[2^(m+1)-1-k] + 2^(m+1)
}
a
# Yosu Yurramendi, Apr 05 2017

(define (A233280 n) (A003188 (A054429 n)))
;; Alternative version, based on entangling even & odd numbers with odious and evil numbers:
(definec (A233280 n) (cond ((< n 2) n) ((even? n) (A000069 (+ 1 (A233280 (/ n 2))))) (else (A001969 (+ 1 (A233280 (/ (- n 1) 2)))))))

a(n) = A003188(A054429(n)).
a(n) = A063946(A003188(n)).
a(n) = A054429(A154436(n)).
a(0)=0, a(1)=1, and otherwise, a(2n) = A000069(1+a(n)), a(2n+1) = A001969(1+a(n)). [A recurrence based on entangling even & odd numbers with odious and evil numbers]
a(n) = A258746(A180201(n)) = A180201(A117120(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

A225850 Inverse of permutation in A167151.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 5, 10, 12, 14, 16, 7, 18, 20, 22, 24, 26, 9, 28, 30, 32, 34, 36, 38, 40, 11, 42, 44, 46, 48, 50, 52, 54, 56, 13, 58, 60, 62, 64, 66, 68, 70, 72, 74, 15, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 17, 96, 98, 100, 102, 104, 106, 108, 110, 112

Offset: 0

Reinhard Zumkeller, May 17 2013

nonn

For n > 0: a(A005228(n)) = 2*n-1 and a(A030124(n)) = 2*n.

For n > 0: A232739(n) = a(A232739(n+1))/2. - Antti Karttunen, Dec 04 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A167151.

Cf. also A005228, A030124, A232739, A232746, A232747, A232749, and also the permutation pair A232751/A232752.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a225850 = fromJust . (`elemIndex` a167151_list)

nmax = 100; A5228 = {1};
Module[{d = 2, k = 1}, Do[While[MemberQ[A5228, d], d++]; k += d; d++; AppendTo[A5228, k], {n, 1, nmax}]];
a46[n_] := For[k = 1, True, k++, If[A5228[[k]] > n, Return[k - 1]]];
a47[n_] := If[n == 1, 1, a46[n] (a46[n] - a46[n - 1])];
a48[n_] := a48[n] = If[n == 1, 0, a48[n-1] + (1 - (a46[n] - a46[n-1]))];
a49[n_] := If[n == 1, 0, a48[n] (a48[n] - a48[n - 1])];
a[n_] := If[n < 3, n, 2 (a47[n] + a49[n]) - (a46[n] - a46[n - 1])];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 09 2021 *)

(define (A225850 n) (if (< n 3) n (- (* 2 (+ (A232747 n) (A232749 n))) (- (A232746 n) (A232746 (- n 1))))))
;; Antti Karttunen, Dec 04 2013

If n < 3, a(n) = n, otherwise a(n) = (2*(A232747(n)+A232749(n))) - (A232746(n)-A232746(n-1)). - Antti Karttunen, Dec 04 2013

cp's OEIS Frontend

A193231 Blue code for n: in binary coding of a polynomial over GF(2), substitute x+1 for x (see Comments for precise definition).

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A005228 Sequence and first differences (A030124) together list all positive numbers exactly once.

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A233275 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with even and odd numbers.

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A233276 a(0)=0, a(1)=1, after which a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).

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A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

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A233278 a(0)=0, a(1)=1, after which a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).

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A232752 Permutation of natural numbers: a(0)=0, a(1)=1, a(2n)=A005228(1+(a(n))), a(2n+1)=A030124(a(n)).

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A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

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