A268823 -id:A268823 - cp's OEIS frontend

A268825 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1+A268823(n-1)).

0, 1, 3, 2, 6, 7, 4, 5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11, 8, 9, 50, 51, 48, 49, 18, 19, 16, 17, 30, 31, 28, 29, 22, 23, 20, 21, 98, 99, 96, 97, 34, 35, 32, 33, 46, 47, 44, 45, 38, 39, 36, 37, 54, 55, 52, 53, 62, 63, 60, 61, 42, 43, 40, 41, 58, 59, 56, 57, 194, 195, 192, 193, 66, 67, 64, 65, 78, 79, 76, 77, 70, 71, 68, 69, 86, 87

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "fourth shifted power" of permutation A268717.

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

Inverse: A268826.
Cf. A268717, A268823, A268827.
Row 4 of array A268820.

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m = A006068[Floor[n/2]]}, 2m + Mod[Mod[n, 2] + Mod[m, 2], 2]]]; A268717[n_]:=If[n<1, 0, A003188[ 1 + A006068[n - 1]]]; A268823[n_]:= If[n<2, n, A268717[1 + A268717[1 + A268717[n - 2]]]]; A268825[n_]:=If[n<1, 0, A268717[1 + A268823[n - 1]]]; Table[A268825[n], {n, 0, 100}] (* Indranil Ghosh, Apr 03 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));
A268823(n) = if(n<2, n, A268717(1 + A268717(1 + A268717(n - 2))));
for(n=0, 100, print1(if(n<1, 0,  A268717(1+A268823(n - 1))),", ")) \\ Indranil Ghosh, Apr 03 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
def A268823(n): return A268717(1 + A268717(1 + A268717(n - 2))) if n>1 else n
def a(n): return A268717(1 + A268823(n - 1)) if n>0 else 0
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 03 2017

(define (A268825 n) (if (zero? n) n (A268717 (+ 1 (A268823 (- n 1))))))

a(0) = 0, and for n >= 1, a(n) = A268717(1+A268823(n-1)).

A268676 a(n) = A101080(n,A268823(3+n)), where A101080(x,y) gives the Hamming distance between binary expansions of x and y.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 19 2016

nonn

It seems that A001969 gives the positions of 1's, while A000069 gives the positions of 3's.

The above observation follows because by definition, a(n) gives the Hamming distance between binary expansions of n and A003188(3+A006068(n)). To see how this leads to the stated claim, consider the illustration "Visualized as a traversal of vertices of a tesseract" in the Wikipedia article "Gray code". Starting from any vertex with either (A) an even, or (B) an odd number of 1-bits, traverse three edges along the red path, to the direction indicated by the arrow, and then note the Hamming distance between the starting and the ending vertex. It is always 1 in case (A), and 3 in case (B), because the position of the flipped bit is given by sequence A007814, with its every other term zero, so in case (A) the third flip cancels the first flip (both toggling the rightmost bit), which leaves only the second bit-flip effective. Note that the properties of a tesseract generalize to those of an infinite dimensional hypercube. - Antti Karttunen, Mar 11 2024

Antti Karttunen, Table of n, a(n) for n = 0..8191
Wikipedia, Gray code.

Row 3 of A268833.

Cf. A000069, A000120, A001969, A003188, A006068, A007814, A010060, A101080, A106400, A268823.

a := n -> 2-(-1)^add(convert(n, base, 2)):
seq(a(n), n = 0 .. 120); # Lorenzo Sauras Altuzarra, Mar 10 2024

A003188(n) = bitxor(n, n>>1);
A006068(n) = { my(s=1, ns); while(1, ns = n >> s; if(0==ns, return(n)); n = bitxor(n, ns); s <<= 1); };
A268676(n) = hammingweight(bitxor(n,A003188(3+A006068(n)))); \\ Antti Karttunen, Mar 11 2024

A268676(n) = 2-((-1)^hammingweight(n)); \\ Antti Karttunen, Mar 11 2024, after Lorenzo Sauras Altuzarra's Maple-code.

(define (A268676 n) (A101080bi n (A268823 (+ 3 n))))
;; Where A101080bi implements the dyadic function A101080(x,y) which gives the Hamming distance between binary expansions of x and y.

a(n) = A101080(n,A268823(3+n)), where A101080(x,y) gives the Hamming distance between binary expansions of x and y.
a(n) = 2 - (-1)^A000120(n) = 2 - A106400(n). - Lorenzo Sauras Altuzarra, Mar 10 2024
a(n) = 1 + 2 * A010060(n). - Joerg Arndt, Mar 11 2024

A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14, 48, 16, 19, 17, 21, 23, 18, 22, 25, 27, 30, 26, 20, 28, 31, 29, 96, 32, 35, 33, 37, 39, 34, 38, 41, 43, 46, 42, 36, 44, 47, 45, 49, 51, 54, 50, 60, 52, 55, 53, 40, 56, 59, 57, 61, 63, 58, 62, 192, 64, 67, 65, 69, 71, 66, 70, 73, 75, 78, 74, 68, 76, 79, 77, 81

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

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

Inverse: A268718.
Cf. A000523, A003188, A003987, A006068, A010060, A066194, A101080, A171977, A268718, A268726, A268727.
Row 1 and column 1 of array A268715 (without the initial zero).
Row 1 of array A268820.
Cf. A092246 (fixed points).
Cf. A268817 ("square" of this permutation).
Cf. A268821 ("shifted square"), A268823 ("shifted cube") and also A268825, A268827 and A268831 ("shifted higher powers").

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := If[n == 0, 0, BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]]; a[n_] := If[n == 0, 0, A003188[1 + A006068[n-1]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)

A003188(n) = bitxor(n, floor(n/2));
A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2});
for(n=0, 100, print1(if(n<1, 0, A003188(1 + A006068(n - 1)))", ")) \\ Indranil Ghosh, Mar 31 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    m = A006068(n//2)
    return 2*m + (n%2 + m%2)%2
def a(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
print([a(n) for n in range(0, 101)]) # Indranil Ghosh, Mar 31 2017

def A268717(n):
    k, m = n-1, n-1>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k+1^ k+1>>1 # Chai Wah Wu, Jun 29 2022

(define (A268717 n) (if (zero? n) n (A003188 (A066194 n))))

a(n) = A003188(A066194(n)) = A003188(1+A006068(n-1)).
Other identities. For all n >= 0:
A101080(n,a(n+1)) = 1. [The Hamming distance between n and a(n+1) is always one.]
A268726(n) = A000523(A003987(n, a(n+1))). [A268726 gives the index of the toggled bit.]
From Alan Michael Gómez Calderón, May 29 2025: (Start)
a(2*n) = (2*n-1) XOR (2-A010060(n-1)) for n >= 1;
a(n) = (A268718(n-1)-1) XOR (A171977(n-1)+1) for n >= 2. (End)

A268820 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = A003188(1+A006068(A(r-1,c-1))) = A268717(1+A(r-1,c-1)), read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 3, 1, 0, 5, 2, 2, 3, 1, 0, 6, 12, 7, 2, 3, 1, 0, 7, 4, 6, 6, 2, 3, 1, 0, 8, 7, 13, 5, 6, 2, 3, 1, 0, 9, 5, 12, 7, 7, 6, 2, 3, 1, 0, 10, 24, 5, 15, 4, 7, 6, 2, 3, 1, 0, 11, 8, 4, 13, 5, 5, 7, 6, 2, 3, 1, 0, 12, 11, 25, 4, 14, 12, 5, 7, 6, 2, 3, 1, 0, 13, 9, 24, 12, 15, 4, 4, 5, 7, 6, 2, 3, 1, 0, 14, 13, 9, 27, 12, 10, 13, 4, 5, 7, 6, 2, 3, 1, 0

Offset: 0

Antti Karttunen, Feb 14 2016

			The top left [0 .. 16] x [0 .. 19] section of the array:
0, 1, 2, 3, 4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
0, 1, 3, 6, 2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14, 48, 16, 19
0, 1, 3, 2, 7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10, 49, 48
0, 1, 3, 2, 6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11, 51
0, 1, 3, 2, 6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
0, 1, 3, 2, 6,  7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24
0, 1, 3, 2, 6,  7,  5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8
0, 1, 3, 2, 6,  7,  5,  4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 11, 10, 25, 24,  9,  8, 21, 20
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10,  9, 11, 27, 25,  8, 24, 23
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  8,  9, 26, 27, 24, 25
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9, 24,  8, 30, 26, 25
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9,  8, 25, 24, 31, 30
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9,  8, 24, 27, 25, 29
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9,  8, 24, 25, 26, 27

Antti Karttunen, Table of n, a(n) for n = 0..32895; the first 256 antidiagonals of array

Cf. A003188, A006068.
Inverses of these permutations can be found in table A268830.
Row 0: A001477, Row 1: A268717, Row 2: A268821, Row 3: A268823, Row 4: A268825, Row 5: A268827, Row 6: A268831, Row 7: A268933.
Rows converge towards A003188, which is also the main diagonal.
Cf. array A268715 (can be extracted from this one).
Cf. array A268833 (shows related Hamming distances with regular patterns).

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m=A006068[Floor[n/2]]}, 2m + Mod[Mod[n,2] + Mod[m, 2], 2]]]; a[r_, 0]:= 0; a[0, c_]:=c; a[r_, c_]:= A003188[1 + A006068[a[r - 1, c - 1]]]; Table[a[c, r - c], {r, 0, 15}, {c, 0, r}] //Flatten (* Indranil Ghosh, Apr 02 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
a(r, c) = if(r==0, c, if(c==0, 0, A003188(1 + A006068(a(r - 1, c - 1)))));
for(r=0, 15, for(c=0, r, print1(a(c, r - c),", "); ); print(); ); \\ Indranil Ghosh, Apr 02 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def a(r, c): return c if r<1 else 0 if c<1 else A003188(1 + A006068(a(r - 1, c - 1)))
for r in range(16):
    print([a(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 02 2017

(define (A268820 n) (A268820bi (A002262 n) (A025581 n)))
(define (A268820bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (A268717 (+ 1 (A268820bi (- row 1) (- col 1)))))))
(define (A268820bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (A003188 (+ 1 (A006068 (A268820bi (- row 1) (- col 1))))))))

For row zero: A(0,k) = k, for column zero: A(n,0) = 0, and in other cases: A(n,k) = A003188(1+A006068(A(n-1,k-1)))
Other identities. For all n >= 0:
A(n,n) = A003188(n).
A(A006068(n),A006068(n)) = n.

A268824 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268822(A268718(n)-1).

Original entry on oeis.org

0, 1, 3, 2, 9, 5, 4, 6, 13, 17, 16, 18, 10, 8, 15, 7, 21, 25, 24, 26, 34, 32, 23, 31, 14, 12, 27, 11, 33, 29, 28, 30, 37, 41, 40, 42, 50, 48, 39, 47, 62, 60, 43, 59, 49, 45, 44, 46, 22, 20, 51, 19, 57, 53, 52, 54, 61, 65, 64, 66, 58, 56, 63, 55, 69, 73, 72, 74, 82, 80, 71, 79, 94, 92, 75, 91, 81, 77, 76, 78, 118, 116, 83, 115, 89, 85

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "third shifted power" of permutation A268718.

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

Inverse: A268823.
Cf. A268718, A268822, A268826.
Row 3 of array A268830.

(define (A268824 n) (if (<= n 1) n (+ 2 (A268718 (+ -1 (A268718 (+ -1 (A268718 n))))))))

a(0) = 0, a(n) = 1 + A268822(A268718(n)-1).

a(0) = 0, a(1) = 1, for n>1: a(n) = 2 + A268718(-1+A268718(-1+A268718(n))).

A268821 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1 + A268717(n-1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 13, 12, 5, 4, 25, 24, 9, 8, 15, 14, 11, 10, 49, 48, 17, 16, 23, 22, 19, 18, 27, 26, 31, 30, 21, 20, 29, 28, 97, 96, 33, 32, 39, 38, 35, 34, 43, 42, 47, 46, 37, 36, 45, 44, 51, 50, 55, 54, 61, 60, 53, 52, 41, 40, 57, 56, 63, 62, 59, 58, 193, 192, 65, 64, 71, 70, 67, 66, 75, 74, 79, 78, 69, 68, 77, 76, 83, 82

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "shifted square" of permutation A268717.

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

Inverse: A268822.
Row 2 of array A268820.
From term a(2) onward (3, 2, 7, 6, ...) also row 3 of A268715.
Cf. A268717, A268823.
Cf. also A101080, A268833.

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m = A006068[Floor[n/2]]}, 2m + Mod[Mod[n, 2] + Mod[m, 2], 2]]]; A268717[n_]:=If[n<1, 0, A003188[1 + A006068[n - 1]]]; Table[If[n<2, n, A268717[1 + A268717[n - 1]]], {n, 0, 100}] (* Indranil Ghosh, Apr 01 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));
for(n=0, 100, print1(if(n<2, n, A268717(1 + A268717(n - 1))), ", ")) \\ Indranil Ghosh, Apr 01 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
def a(n): return A268717(1 + A268717(n-1)) if n>0 else 0
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 01 2017

(define (A268821 n) (if (zero? n) n (A268717 (+ 1 (A268717 (- n 1))))))

a(0) = 0, for n >= 1, a(n) = A268717(1 + A268717(n-1)).
Other identities. For all n >= 0:
A101080(n, a(n+2)) = 2.

A268827 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1+A268825(n-1)).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 12, 4, 10, 14, 13, 15, 30, 26, 25, 27, 11, 9, 24, 8, 54, 50, 49, 51, 19, 17, 48, 16, 31, 29, 20, 28, 18, 22, 21, 23, 102, 98, 97, 99, 35, 33, 96, 32, 47, 45, 36, 44, 34, 38, 37, 39, 55, 53, 60, 52, 58, 62, 61, 63, 46, 42, 41, 43, 59, 57, 40, 56, 198, 194, 193, 195, 67, 65, 192, 64, 79, 77, 68, 76, 66, 70, 69, 71, 87

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "fifth shifted power" of permutation A268717.

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

Inverse: A268828.
Cf. A268717, A268825, A268831.
Row 5 of array A268820.

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m = A006068[Floor[n/2]]}, 2m + Mod[Mod[n, 2] + Mod[m, 2], 2]]]; A268717[n_]:=If[n<1, 0, A003188[ 1 + A006068[n - 1]]]; A268823[n_]:= If[n<2, n, A268717[1 + A268717[1 + A268717[n - 2]]]]; A268825[n_]:=If[n<1, 0, A268717[1 + A268823[n - 1]]];  A268827[n_]:=If[n<1, 0, A268717[1 + A268825[n - 1]]]; Table[A268827[n], {n, 0, 100}] (* Indranil Ghosh, Apr 03 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));
A268823(n) = if(n<2, n, A268717(1 + A268717(1 + A268717(n - 2))));
A268825(n) = if(n<1, 0, A268717(1+A268823(n - 1)));
for(n=0, 100, print1(if(n<1, 0,  A268717(1+A268825(n - 1))),", ")) \\ Indranil Ghosh, Apr 03 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
def A268823(n): return A268717(1 + A268717(1 + A268717(n - 2))) if n>1 else n
def A268825(n): return A268717(1 + A268823(n - 1)) if n>0 else 0
def a(n): return A268717(1 + A268825(n - 1)) if n>0 else 0
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 03 2017

(define (A268827 n) (if (zero? n) n (A268717 (+ 1 (A268825 (- n 1))))))

a(0) = 0, for n >= 1, a(n) = A268717(1+A268825(n-1)).

A268831 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1+A268827(n-1)).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26, 9, 8, 25, 24, 55, 54, 51, 50, 17, 16, 49, 48, 29, 28, 21, 20, 19, 18, 23, 22, 103, 102, 99, 98, 33, 32, 97, 96, 45, 44, 37, 36, 35, 34, 39, 38, 53, 52, 61, 60, 59, 58, 63, 62, 47, 46, 43, 42, 57, 56, 41, 40, 199, 198, 195, 194, 65, 64, 193, 192, 77

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The sixth "shifted power" of A268717.

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

Inverse: A268832.
Row 6 of A268820.
Cf. A003188, A006068, A268717, A268827.

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m = A006068[Floor[n/2]]}, 2m + Mod[Mod[n, 2] + Mod[m, 2], 2]]]; A268717[n_]:=If[n<1, 0, A003188[ 1 + A006068[n - 1]]]; A268823[n_]:= If[n<2, n, A268717[1 + A268717[1 + A268717[n - 2]]]]; A268825[n_]:=If[n<1, 0, A268717[1 + A268823[n - 1]]];  A268827[n_]:=If[n<1, 0, A268717[1 + A268825[n - 1]]]; A268831[n_]:=If[n<1, 0, A268717[1 + A268827[n - 1]]];Table[A268831[n], {n, 0, 100}] (* Indranil Ghosh, Apr 03 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));
A268823(n) = if(n<2, n, A268717(1 + A268717(1 + A268717(n - 2))));
A268825(n) = if(n<1, 0,  A268717(1+A268823(n - 1)));
A268827(n) = if(n<1, 0,  A268717(1+A268825(n - 1)));
for(n=0, 100, print1(if(n<1, 0,  A268717(1+A268827(n - 1))),", ")) \\ Indranil Ghosh, Apr 03 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
def A268823(n): return A268717(1 + A268717(1 + A268717(n - 2))) if n>1 else n
def A268825(n): return A268717(1 + A268823(n - 1)) if n>0 else 0
def A268827(n): return A268717(1 + A268825(n - 1)) if n>0 else 0
def a(n): return A268717(1 + A268827(n - 1)) if n>0 else 0
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 03 2017

(define (A268831 n) (if (zero? n) n (A268717 (+ 1 (A268827 (- n 1))))))

a(0) = 0, for n >= 1, a(n) = A268717(1+A268827(n-1)).

A268673 a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

Original entry on oeis.org

0, 1, 5, 29, 45, 53, 77, 85, 101, 125, 141, 149, 165, 189, 197, 221, 237, 245, 269, 277, 293, 317, 325, 349, 365, 373, 389, 413, 429, 437, 461, 469, 485, 509, 525, 533, 549, 573, 581, 605, 621, 629, 645, 669, 685, 693, 717, 725, 741, 765, 773, 797, 813, 821, 845, 853, 869, 893, 909, 917, 933, 957, 965, 989, 1005

Offset: 0

Antti Karttunen, Feb 19 2016

nonn

Seems to be also the fixed points of permutations A268823 and A268824.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A268823, A268824.
Cf. also A092246, A268717.
Many (but not all) terms of A013710 seem to be included.

Join[{0, 1}, 1 + 4 Select[Range[1, 251, 2], OddQ[Total[IntegerDigits[#, 2]]]&]] (* Jean-François Alcover, Mar 15 2016 *)

def A268673(n): return (((m:=n-2)<<4)+(13 if m.bit_count()&1 else 5)) if n>1 else n # Chai Wah Wu, Mar 03 2023

(define (A268673 n) (if (<= n 1) n (+ 1 (* 4 (A092246 (- n 1))))))

a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

cp's OEIS Frontend

A268825 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1+A268823(n-1)).

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A268676 a(n) = A101080(n,A268823(3+n)), where A101080(x,y) gives the Hamming distance between binary expansions of x and y.

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A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

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A268820 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = A003188(1+A006068(A(r-1,c-1))) = A268717(1+A(r-1,c-1)), read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A268824 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268822(A268718(n)-1).

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A268821 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1 + A268717(n-1)).

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A268827 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1+A268825(n-1)).

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