A268825 -id:A268825

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A268827

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

+20
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The "fifth shifted power" of permutation A268717.

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

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

(PARI) 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

(Python)

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

CROSSREFS

Inverse: A268828.

Cf. A268717, A268825, A268831.

Row 5 of array A268820.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 14 2016

STATUS

approved

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.

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

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.]

MATHEMATICA

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 *)

PROG

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

(PARI) 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

(Python)

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

(Python)

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

CROSSREFS

Inverse: A268718.

Cf. A000523, A003188, A003987, A006068, A066194, A101080, 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").

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 12 2016

STATUS

approved

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), ...

+10
13

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

FORMULA

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.

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(Scheme)

(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))))))))

(PARI) 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

(Python)

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

CROSSREFS

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).

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Feb 14 2016

STATUS

approved

A268823

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The "third shifted power" of permutation A268717.

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

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

PROG

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

(PARI) 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

(Python)

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

CROSSREFS

Inverse: A268824.

Cf. A003188, A006068, A268717, A268821, A268825, A268676.

Row 3 of array A268820.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 14 2016

STATUS

approved

A268826

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The "fourth shifted power" of permutation A268718.

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

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

PROG

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

CROSSREFS

Inverse: A268825.

Cf. A268718, A268824, A268828.

Row 4 of array A268830.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 14 2016

STATUS

approved

A268831

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The sixth "shifted power" of A268717.

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

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

(PARI) 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

(Python)

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

CROSSREFS

Inverse: A268832.

Row 6 of A268820.

Cf. A003188, A006068, A268717, A268827.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 14 2016

STATUS

approved

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