A268821 -id:A268821 - cp's OEIS frontend

A268823 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1 + A268821(n-1)).

0, 1, 3, 2, 6, 5, 7, 15, 13, 4, 12, 27, 25, 8, 24, 14, 10, 9, 11, 51, 49, 16, 48, 22, 18, 17, 19, 26, 30, 29, 31, 23, 21, 28, 20, 99, 97, 32, 96, 38, 34, 33, 35, 42, 46, 45, 47, 39, 37, 44, 36, 50, 54, 53, 55, 63, 61, 52, 60, 43, 41, 56, 40, 62, 58, 57, 59, 195, 193, 64, 192, 70, 66, 65, 67, 74, 78, 77, 79, 71, 69, 76, 68, 82, 86, 85

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "third shifted power" 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: A268824.
Cf. A003188, A006068, A268717, A268821, A268825, A268676.
Row 3 of array A268820.

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});
A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));
for(n=0, 100, print1(if(n<2, n, A268717(1 + A268717(1 + A268717(n - 2)))),", ")) \\ Indranil Ghosh, Mar 31 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(1 + A268717(n - 2))) if n>1 else n
print([a(n) for n in range(101)]) # Indranil Ghosh, Mar 31 2017

(define (A268823 n) (if (<= n 1) n (A268717 (+ 1 (A268717 (+ 1 (A268717 (- n 2))))))))

a(0), for n >= 1, a(n) = A268717(1 + A268821(n-1)).
a(0) = 0, a(1) = 1, and for n > 1, a(n) = A268717(1 + A268717(1 + A268717(n-2))).
For n >= 3, a(n) = A003188(3+A006068(n-3)). - Antti Karttunen, 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.

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

Original entry on oeis.org

0, 1, 3, 2, 9, 8, 5, 4, 13, 12, 17, 16, 7, 6, 15, 14, 21, 20, 25, 24, 31, 30, 23, 22, 11, 10, 27, 26, 33, 32, 29, 28, 37, 36, 41, 40, 47, 46, 39, 38, 59, 58, 43, 42, 49, 48, 45, 44, 19, 18, 51, 50, 57, 56, 53, 52, 61, 60, 65, 64, 55, 54, 63, 62, 69, 68, 73, 72, 79, 78, 71, 70, 91, 90, 75, 74, 81, 80, 77, 76, 115, 114, 83, 82

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "shifted square" 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: A268821.
Cf. A268718, A268824.
Row 2 of array A268830.

(define (A268822 n) (if (zero? n) n (+ 1 (A268718 (+ -1 (A268718 n))))))

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

A268817 Permutation of nonnegative integers: a(n) = A268717(A268717(n)).

Original entry on oeis.org

0, 1, 6, 4, 3, 9, 2, 7, 12, 22, 5, 11, 24, 13, 10, 8, 15, 45, 14, 19, 48, 21, 18, 16, 23, 25, 30, 28, 27, 17, 26, 31, 20, 94, 29, 35, 96, 37, 34, 32, 39, 41, 46, 44, 43, 33, 42, 47, 36, 49, 54, 52, 51, 57, 50, 55, 60, 38, 53, 59, 40, 61, 58, 56, 63, 189, 62, 67, 192, 69, 66, 64, 71, 73, 78, 76, 75, 65, 74, 79, 68, 81

Offset: 0

Antti Karttunen, Feb 14 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: A268818.
Cf. A268717.
Cf. also A268821.

(define (A268817 n) (A268717 (A268717 n)))

a(n) = A268717(A268717(n)).

cp's OEIS Frontend

A268823 Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1 + A268821(n-1)).

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

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A268817 Permutation of nonnegative integers: a(n) = A268717(A268717(n)).

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