A268718 -id:A268718 - cp's OEIS frontend

A268830 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1,A268718(c)-1) = 1 + A(r-1, A003188(A006068(c)-1)), read by descending antidiagonals.

0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 2, 3, 1, 0, 5, 6, 2, 3, 1, 0, 6, 8, 9, 2, 3, 1, 0, 7, 3, 8, 9, 2, 3, 1, 0, 8, 7, 5, 5, 6, 2, 3, 1, 0, 9, 10, 4, 4, 7, 8, 2, 3, 1, 0, 10, 12, 13, 6, 4, 6, 7, 2, 3, 1, 0, 11, 15, 12, 13, 5, 4, 6, 7, 2, 3, 1, 0, 12, 11, 17, 17, 18, 5, 4, 6, 7, 2, 3, 1, 0, 13, 5, 16, 16, 19, 20, 5, 4, 6, 7, 2, 3, 1, 0, 14, 13, 7, 18, 16, 18, 19, 5, 4, 6, 7, 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, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11,  5, 13, 16, 14, 18, 20, 23, 19
0, 1, 3, 2, 9, 8, 5, 4, 13, 12, 17, 16,  7,  6, 15, 14, 21, 20, 25, 24
0, 1, 3, 2, 9, 5, 4, 6, 13, 17, 16, 18, 10,  8, 15,  7, 21, 25, 24, 26
0, 1, 3, 2, 6, 7, 4, 5, 18, 19, 16, 17, 10, 11,  8,  9, 26, 27, 24, 25
0, 1, 3, 2, 8, 6, 4, 5, 20, 18,  9, 17,  7, 11, 10, 12, 28, 26, 33, 25
0, 1, 3, 2, 7, 6, 4, 5, 19, 18, 11, 10,  9,  8, 13, 12, 27, 26, 35, 34
0, 1, 3, 2, 7, 6, 4, 5, 19, 11, 14, 12,  8, 10, 13,  9, 27, 35, 38, 36
0, 1, 3, 2, 7, 6, 4, 5, 12, 13, 14, 15,  8,  9, 10, 11, 36, 37, 38, 39
0, 1, 3, 2, 7, 6, 4, 5, 14, 16, 11, 15,  8,  9, 12, 10, 38, 40, 35, 39
0, 1, 3, 2, 7, 6, 4, 5, 17, 16, 13, 12,  8,  9, 11, 10, 41, 40, 37, 36
0, 1, 3, 2, 7, 6, 4, 5, 17, 13, 12, 14,  8,  9, 11, 10, 41, 37, 36, 38
0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 12, 13,  8,  9, 11, 10, 38, 39, 36, 37
0, 1, 3, 2, 7, 6, 4, 5, 16, 14, 12, 13,  8,  9, 11, 10, 40, 38, 21, 37
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 39, 38, 23, 22
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 39, 23, 26, 24
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 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 A268820.
Row 0: A001477, Row 1: A268718, Row 2: A268822, Row 3: A268824, Row 4: A268826, Row 5: A268828, Row 6: A268832, Row 7: A268934.
Rows converge towards A006068.

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a278618(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1)
def A(r, c): return c if r==0 else 0 if c==0 else 1 + A(r - 1, a278618(c) - 1)
for r in range(21): print([A(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268830 n) (A268830bi (A002262 n) (A025581 n))) ;; o=0: Square array of shifted powers of A268718.
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (- (A268718 col) 1))))))
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (A003188 (+ -1 (A006068 col))))))))

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))).

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)).

A268826 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268824(A268718(n)-1).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 18, 19, 16, 17, 10, 11, 8, 9, 26, 27, 24, 25, 34, 35, 32, 33, 14, 15, 12, 13, 30, 31, 28, 29, 42, 43, 40, 41, 50, 51, 48, 49, 62, 63, 60, 61, 46, 47, 44, 45, 22, 23, 20, 21, 54, 55, 52, 53, 66, 67, 64, 65, 58, 59, 56, 57, 74, 75, 72, 73, 82, 83, 80, 81, 94, 95, 92, 93, 78, 79, 76, 77, 118, 119, 116, 117, 86, 87

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "fourth 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: A268825.
Cf. A268718, A268824, A268828.
Row 4 of array A268830.

(define (A268826 n) (if (zero? n) n (+ 1 (A268824 (+ -1 (A268718 n))))))

a(0) = 0, and for n >= 1, a(n) = 1 + A268824(A268718(n)-1).

A268828 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268826(A268718(n)-1).

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 4, 5, 20, 18, 9, 17, 7, 11, 10, 12, 28, 26, 33, 25, 31, 35, 34, 36, 19, 15, 14, 16, 32, 30, 13, 29, 44, 42, 49, 41, 47, 51, 50, 52, 67, 63, 62, 64, 48, 46, 61, 45, 27, 23, 22, 24, 56, 54, 21, 53, 68, 66, 57, 65, 55, 59, 58, 60, 76, 74, 81, 73, 79, 83, 82, 84, 99, 95, 94, 96, 80, 78, 93, 77, 123, 119, 118, 120, 88, 86

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The "fifth 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: A268827.
Cf. A268718, A268826, A268832.
Row 5 of array A268830.

(define (A268828 n) (if (zero? n) n (+ 1 (A268826 (+ -1 (A268718 n))))))

a(0) = 0, and for n >= 1, a(n) = 1 + A268826(A268718(n)-1).

A268832 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268828(A268718(n)-1).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 19, 18, 11, 10, 9, 8, 13, 12, 27, 26, 35, 34, 33, 32, 37, 36, 21, 20, 17, 16, 31, 30, 15, 14, 43, 42, 51, 50, 49, 48, 53, 52, 69, 68, 65, 64, 47, 46, 63, 62, 29, 28, 25, 24, 55, 54, 23, 22, 67, 66, 59, 58, 57, 56, 61, 60, 75, 74, 83, 82, 81, 80, 85, 84, 101, 100, 97, 96, 79, 78, 95, 94, 125, 124, 121, 120

Offset: 0

Antti Karttunen, Feb 14 2016

nonn

The sixth "shifted power" of A268718.

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

Inverse: A268831.
Row 6 of A268830.
Cf. A003188, A006068, A268718, A268828.

(define (A268832 n) (if (zero? n) n (+ 1 (A268828 (+ -1 (A268718 n))))))

a(0) = 0, and for n >= 1, a(n) = 1 + A268828(A268718(n)-1).

A268818 Permutation of nonnegative integers: a(n) = A268718(A268718(n)).

Original entry on oeis.org

0, 1, 6, 4, 3, 10, 2, 7, 15, 5, 14, 11, 8, 13, 18, 16, 23, 29, 22, 19, 32, 21, 9, 24, 12, 25, 30, 28, 27, 34, 26, 31, 39, 45, 38, 35, 48, 37, 57, 40, 60, 41, 46, 44, 43, 17, 42, 47, 20, 49, 54, 52, 51, 58, 50, 55, 63, 53, 62, 59, 56, 61, 66, 64, 71, 77, 70, 67, 80, 69, 89, 72, 92, 73, 78, 76, 75, 113, 74, 79, 116, 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: A268817.
Cf. A268718.
Cf. also A268822.

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a278618(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1)
def a(n): return a278618(a278618(n)) # Indranil Ghosh, Jun 07 2017

(define (A268818 n) (A268718 (A268718 n)))

a(n) = A268718(A268718(n)).

A268934 Permutation of nonnegative integers: a(0) = 0, for n >= 1, a(n) = 1 + A268832(A268718(n)-1).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 19, 11, 14, 12, 8, 10, 13, 9, 27, 35, 38, 36, 32, 34, 37, 33, 20, 22, 17, 21, 31, 15, 18, 16, 43, 51, 54, 52, 48, 50, 53, 49, 68, 70, 65, 69, 47, 63, 66, 64, 28, 30, 25, 29, 55, 23, 26, 24, 67, 59, 62, 60, 56, 58, 61, 57, 75, 83, 86, 84, 80, 82, 85, 81, 100, 102, 97, 101, 79, 95, 98, 96, 124

Offset: 0

Antti Karttunen, Feb 16 2016

nonn

The seventh "shifted power" of A268718.

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

Cf. A003188, A006068, A268718, A268832.
Inverse: A268933.
Row 7 of A268830.

(define (A268934 n) (if (zero? n) n (+ 1 (A268832 (+ -1 (A268718 n))))))

a(0) = 0, for n >= 1, a(n) = 1 + A268832(A268718(n)-1).

A092246 Odd "odious" numbers (A000069).

Original entry on oeis.org

1, 7, 11, 13, 19, 21, 25, 31, 35, 37, 41, 47, 49, 55, 59, 61, 67, 69, 73, 79, 81, 87, 91, 93, 97, 103, 107, 109, 115, 117, 121, 127, 131, 133, 137, 143, 145, 151, 155, 157, 161, 167, 171, 173, 179, 181, 185, 191, 193, 199, 203, 205, 211, 213, 217, 223, 227, 229, 233

Offset: 1

Benoit Cloitre, Feb 23 2004

In other words, numbers having a binary representation ending in 1, and an odd number of 1's overall. It follows that by decrementing an odd odious number, one gets an even evil number (A125592). - Ralf Stephan, Aug 27 2013
The members of the sequence may be called primitive odious numbers because every odious number is a power of 2 times one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - T. D. Noe, Jun 06 2007
From Gary W. Adamson, Apr 06 2010: (Start)
a(n) = A026147(n)-th odd number, where A026147 = (1, 4, 6, 7, 10, 11, ...); e.g.,
         n: 1   2   3   4   5   6   7   8   9  10  11
  n-th odd: 1   3   5   7   9  11  13  15  17  19  21
      a(n): 1           7      11  13          19  21
etc. (End)
Numbers m, such that when merge-sorting lists of length m, the maximal number of comparisons is even: A003071(a(n)) = A230720(n). - Reinhard Zumkeller, Oct 28 2013
Fixed points of permutation pair A268717/A268718. - Antti Karttunen, Feb 29 2016

Cf. A129771, A026147.
Cf. A230709 (complement).
Cf. A268717, A268718, A268673.

a092246 n = a092246_list !! (n - 1)
a092246_list = filter odd a000069_list
-- Reinhard Zumkeller, Oct 28 2013

Table[If[n < 1, 0, 2 n - 1 - Mod[First@ DigitCount[n - 1, 2], 2]], {n, 120}] /. n_ /; EvenQ@ n -> Nothing (* Michael De Vlieger, Feb 29 2016 *)
Select[Range[1, 1001, 2], OddQ[Total[IntegerDigits[#, 2]]]&] (* Jean-François Alcover, Mar 15 2016 *)

is(n)=n%2&&hammingweight(n)%2 \\ Charles R Greathouse IV, Mar 21 2013

a(n)=4*n-if(hammingweight(n-1)%2,1,3) \\ Charles R Greathouse IV, Mar 22 2013

def A092246(n): return (n<<2)-(1 if (n-1).bit_count()&1 else 3) # Chai Wah Wu, Mar 03 2023

a(n) = 4*n + 2*A010060(n-1) - 3;

a(n) = 2*A001969(n-1) + 1.

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)

cp's OEIS Frontend

A268830 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1,A268718(c)-1) = 1 + A(r-1, A003188(A006068(c)-1)), read by descending antidiagonals.

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

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

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A268826 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268824(A268718(n)-1).

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A268828 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268826(A268718(n)-1).

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A268832 Permutation of nonnegative integers: a(0) = 0, a(n) = 1 + A268828(A268718(n)-1).

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A268818 Permutation of nonnegative integers: a(n) = A268718(A268718(n)).

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A268934 Permutation of nonnegative integers: a(0) = 0, for n >= 1, a(n) = 1 + A268832(A268718(n)-1).

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