A268715 -id:A268715 - cp's OEIS frontend

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

0, 2, 7, 18, 17, 48, 56, 80, 67, 122, 136, 194, 204, 268, 281, 328, 291, 378, 396, 498, 510, 640, 675, 792, 790, 886, 965, 1098, 1093, 1208, 1248, 1344, 1227, 1378, 1356, 1530, 1538, 1792, 1815, 2016, 2008, 2218, 2339, 2602, 2619, 2892, 2970, 3208, 3150, 3294, 3385, 3586, 3691, 4012, 4174, 4440, 4367, 4554, 4644

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A003188, A006068.

Cf. also A268720, A268836.

(define (A268837 n) (add (lambda (k) (A003188 (+ (A006068 k) (A006068 (- n k))))) 0 n))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

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.

A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 9, 9, 9, 9, 9, 7, 8, 10, 8, 8, 8, 8, 10, 8, 9, 7, 15, 7, 7, 7, 15, 7, 9, 10, 8, 10, 12, 10, 10, 12, 10, 8, 10, 11, 15, 7, 15, 25, 15, 25, 15, 7, 15, 11, 12, 14, 12, 10, 12, 18, 18, 12, 10, 12, 14, 12, 13, 25, 21, 25, 15, 25, 11, 25, 15, 25, 21, 25, 13

Offset: 1

Antti Karttunen, Feb 13 2021

Considered as a binary operation on the positive integers, A(x, y) returns the term of the Doudna-sequence from the position that is the sum of the positions of x and y in the same sequence. (This is based on giving the Doudna-sequence an offset of 0, rather than 1 as used in A005940.) - Peter Munn, Feb 14 2021

			The top left 16x16 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12,  13,  14,  15, 16,
   2,  3,  4,  5,  6,  9, 10,  7,  8, 15,  14,  25,  22,  21,  12, 11,
   3,  4,  5,  6,  9,  8, 15, 10,  7, 12,  21,  18,  33,  20,  25, 14,
   4,  5,  6,  9,  8,  7, 12, 15, 10, 25,  20,  27,  28,  35,  18, 21,
   5,  6,  9,  8,  7, 10, 25, 12, 15, 18,  35,  16,  55,  30,  27, 20,
   6,  9,  8,  7, 10, 15, 18, 25, 12, 27,  30,  11,  42,  45,  16, 35,
   7, 10, 15, 12, 25, 18, 11, 16, 27, 14,  49,  20,  77,  50,  21, 24,
   8,  7, 10, 15, 12, 25, 16, 27, 18, 11,  24,  21,  40,  49,  14, 45,
   9,  8,  7, 10, 15, 12, 27, 18, 25, 16,  45,  14,  63,  24,  11, 30,
  10, 15, 12, 25, 18, 27, 14, 11, 16, 21,  50,  35,  70,  75,  20, 49,
  11, 14, 21, 20, 35, 30, 49, 24, 45, 50,  13,  36, 121,  22,  75, 32,
  12, 25, 18, 27, 16, 11, 20, 21, 14, 35,  36,  45,  60, 125,  30, 75,
  13, 22, 33, 28, 55, 42, 77, 40, 63, 70, 121,  60,  17,  98, 105, 48,
  14, 21, 20, 35, 30, 45, 50, 49, 24, 75,  22, 125,  98,  33,  36, 13,
  15, 12, 25, 18, 27, 16, 21, 14, 11, 20,  75,  30, 105,  36,  35, 50,
  16, 11, 14, 21, 20, 35, 24, 45, 30, 49,  32,  75,  48,  13,  50, 81,

Cf. A341511 (the lower triangular section).
Cf. A003961 (main diagonal), A329603 (skewed diagonal).
Cf. A297165 (row 2 and column 2, when started from its term a(1)).
Cf. also A003056, A268715, A341515, A341520, A348041.

up_to = 105;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341510sq(n,k) = A005940(1+A156552(n)+A156552(k));
A341510list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341510sq(col,(a-(col-1))))); (v); };
v341510 = A341510list(up_to);
A341510(n) = v341510[n];

A(n, k) = A(k, n) = A005940(1 + A156552(n) + A156552(k)).
A(n, n) = A003961(n).
A(n, 2*n) = A(2*n, n) = A329603(n).
A(n, 2) = A(2, n) = A297165(n).

A268725 Square array A(i,j) = A003188(A006068(i) * A006068(j)), read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 2, 3, 13, 3, 4, 5, 5, 4, 5, 31, 6, 31, 5, 6, 27, 9, 9, 27, 6, 7, 10, 10, 41, 10, 10, 7, 8, 8, 12, 63, 63, 12, 8, 8, 9, 59, 15, 18, 54, 18, 15, 59, 9, 10, 63, 17, 50, 20, 20, 50, 17, 63, 10, 11, 54, 18, 93, 17, 24, 17, 93, 18, 54, 11, 12, 52, 20, 83, 119, 30, 30, 119, 83, 20, 52, 12, 13, 20, 23, 126, 126, 34, 21, 34, 126, 126, 23, 20, 13

Offset: 1

Antti Karttunen, Feb 13 2016

			The top left [1 .. 15] x [1 .. 15] section of the array:
   1,  2,  3,   4,   5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15
   2, 13,  5,  31,  27, 10,   8,  59,  63,  54,  52,  20,  22,  49,  17
   3,  5,  6,   9,  10, 12,  15,  17,  18,  20,  23,  24,  27,  29,  30
   4, 31,  9,  41,  63, 18,  50,  93,  83, 126, 118,  36,  32, 107, 101
   5, 27, 10,  63,  54, 20,  17, 119, 126, 108, 105,  40,  45,  99,  34
   6, 10, 12,  18,  20, 24,  30,  34,  36,  40,  46,  48,  54,  58,  60
   7,  8, 15,  50,  17, 30,  21, 110, 101,  34,  97,  60,  59,  44,  43
   8, 59, 17,  93, 119, 34, 110, 145, 187, 238, 162,  68, 196, 247, 221
   9, 63, 18,  83, 126, 36, 101, 187, 166, 252, 237,  72,  65, 215, 202
  10, 54, 20, 126, 108, 40,  34, 238, 252, 216, 210,  80,  90, 198,  68
  11, 52, 23, 118, 105, 46,  97, 162, 237, 210, 253,  92,  79, 200, 195
  12, 20, 24,  36,  40, 48,  60,  68,  72,  80,  92,  96, 108, 116, 120
  13, 22, 27,  32,  45, 54,  59, 196,  65,  90,  79, 108, 121,  82, 119
  14, 49, 29, 107,  99, 58,  44, 247, 215, 198, 200, 116,  82,  69,  89
  15, 17, 30, 101,  34, 60,  43, 221, 202,  68, 195, 120, 119,  89,  86

Antti Karttunen, Table of n, a(n) for n = 1..15051; the first 173 antidiagonals of the array

Cf. A003188, A006068, A268724.
Cf. A268723 (main diagonal).
Cf. A268722 (row 2 and column 2).
Cf. A001969 (row 3 and column 3).
Cf. also A268715.

(define (A268725 n) (A268725bi (A002260 n) (A004736 n)))
(define (A268725bi row col) (A003188 (* (A006068 row) (A006068 col))))

A(i,j) = A003188(A006068(i) * A006068(j)).

A(i,j) = A003188(A268724(i,j)).

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.

A268714 Square array A(i,j) = A006068(i) + A006068(j), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 2, 4, 4, 2, 7, 3, 6, 3, 7, 6, 8, 5, 5, 8, 6, 4, 7, 10, 4, 10, 7, 4, 5, 5, 9, 9, 9, 9, 5, 5, 15, 6, 7, 8, 14, 8, 7, 6, 15, 14, 16, 8, 6, 13, 13, 6, 8, 16, 14, 12, 15, 18, 7, 11, 12, 11, 7, 18, 15, 12, 13, 13, 17, 17, 12, 10, 10, 12, 17, 17, 13, 13, 8, 14, 15, 16, 22, 11, 8, 11, 22, 16, 15, 14, 8, 9, 9, 16, 14, 21, 21, 9, 9, 21, 21, 14, 16, 9, 9

Offset: 0

Antti Karttunen, Feb 12 2016

			The top left [0 .. 15] x [0 .. 15] section of the array:
   0,  1,  3,  2,  7,  6,  4,  5, 15, 14, 12, 13,  8,  9, 11, 10
   1,  2,  4,  3,  8,  7,  5,  6, 16, 15, 13, 14,  9, 10, 12, 11
   3,  4,  6,  5, 10,  9,  7,  8, 18, 17, 15, 16, 11, 12, 14, 13
   2,  3,  5,  4,  9,  8,  6,  7, 17, 16, 14, 15, 10, 11, 13, 12
   7,  8, 10,  9, 14, 13, 11, 12, 22, 21, 19, 20, 15, 16, 18, 17
   6,  7,  9,  8, 13, 12, 10, 11, 21, 20, 18, 19, 14, 15, 17, 16
   4,  5,  7,  6, 11, 10,  8,  9, 19, 18, 16, 17, 12, 13, 15, 14
   5,  6,  8,  7, 12, 11,  9, 10, 20, 19, 17, 18, 13, 14, 16, 15
  15, 16, 18, 17, 22, 21, 19, 20, 30, 29, 27, 28, 23, 24, 26, 25
  14, 15, 17, 16, 21, 20, 18, 19, 29, 28, 26, 27, 22, 23, 25, 24
  12, 13, 15, 14, 19, 18, 16, 17, 27, 26, 24, 25, 20, 21, 23, 22
  13, 14, 16, 15, 20, 19, 17, 18, 28, 27, 25, 26, 21, 22, 24, 23
   8,  9, 11, 10, 15, 14, 12, 13, 23, 22, 20, 21, 16, 17, 19, 18
   9, 10, 12, 11, 16, 15, 13, 14, 24, 23, 21, 22, 17, 18, 20, 19
  11, 12, 14, 13, 18, 17, 15, 16, 26, 25, 23, 24, 19, 20, 22, 21
  10, 11, 13, 12, 17, 16, 14, 15, 25, 24, 22, 23, 18, 19, 21, 20

Antti Karttunen, Table of n, a(n) for n = 0..15050; the first 173 antidiagonals of the array

Cf. A003188, A268715.
Cf. A006068 (row 0, column 0).
Cf. A066194 (row 1, column 1).
Cf. A268716 (main diagonal).
Cf. also A268724.

A006068[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Log[2, n]}]; A006068[0] = 0; A[i_, j_] := A006068[i] + A006068[j]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

\\ Produces the triangle when the array is read by antidiagonals
a(n) = if(n<2, n, 2*a(floor(n/2)) + (n%2 + a(floor(n/2))%2)%2); /* A006068 */
T(i,j) = a(i) + a(j);
for(i=0, 13, for(j=0, i, print1(T(i - j, j),", "););print();); \\ Indranil Ghosh, Mar 23 2017

# Produces the triangle when the array is read by antidiagonals
def A006068(n):
    return n if n<2 else 2*A006068(n//2) + (n%2 + A006068(n//2)%2)%2
def T(i,j): return A006068(i) + A006068(j)
for i in range(14):
    print([T(i - j, j) for j in range(i + 1)]) # Indranil Ghosh, Mar 23 2017

(define (A268714 n) (A268714bi (A002262 n) (A025581 n)))
(define (A268714bi row col) (+ (A006068 row) (A006068 col)))

A(i,j) = A006068(i) + A006068(j).

A(i,j) = A006068(A268715(i,j)). - Corrected Mar 23 2017

A268719 Triangular table T(n>=0,k=0..n) = A003188(A006068(n) + A006068(k)), read by rows as A(0,0), A(1,0), A(1,1), A(2,0), A(2,1), A(2,2), ...

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 3, 2, 7, 6, 4, 12, 15, 13, 9, 5, 4, 13, 12, 11, 10, 6, 7, 4, 5, 14, 15, 12, 7, 5, 12, 4, 10, 14, 13, 15, 8, 24, 27, 25, 29, 31, 26, 30, 17, 9, 8, 25, 24, 31, 30, 27, 26, 19, 18, 10, 11, 8, 9, 26, 27, 24, 25, 22, 23, 20, 11, 9, 24, 8, 30, 26, 25, 27, 18, 22, 21, 23, 12, 13, 14, 15, 8, 9, 10, 11, 28, 29, 30, 31, 24

Offset: 0

Antti Karttunen, Feb 13 2016

			The first fifteen rows of the triangle:
                             0
                           1   3
                         2   6   5
                       3   2   7   6
                     4  12  15  13   9
                   5   4  13  12  11  10
                 6   7   4   5  14  15  12
               7   5  12   4  10  14  13  15
             8  24  27  25  29  31  26  30  17
           9   8  25  24  31  30  27  26  19  18
        10  11   8   9  26  27  24  25  22  23  20
      11   9  24   8  30  26  25  27  18  22  21  23
    12  13  14  15   8   9  10  11  28  29  30  31  24
  13  15  10  14  24   8  11   9  20  28  31  29  25  27
14  10   9  11  27  25   8  24  23  21  28  20  26  30  29

Antti Karttunen, Table of n, a(n) for n = 0..15050; rows 0 .. 172 of the triangular table

Cf. A003188, A006068.
Cf. A002262, A003056, A268715.
Cf. A001477 (left edge), A001969 (right edge).
Cf. A268720 (row sums).

a88[n_] := BitXor[n, Floor[n/2]];
a68[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}];
a68[0] = 0;
T[n_, k_] := a88[a68[n] + a68[k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

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 T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268719 n) (A268715bi (A003056 n) (A002262 n)))

T(n,k) = A003188(A006068(n) + A006068(k)).

a(n) = A268715(A003056(n), A002262(n)). [As a linear sequence.]

A268720 Row sums of A268719: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

Original entry on oeis.org

0, 4, 13, 18, 53, 55, 63, 80, 217, 217, 205, 244, 234, 264, 305, 328, 881, 877, 841, 916, 790, 864, 977, 988, 900, 956, 1021, 1070, 1197, 1235, 1267, 1344, 3553, 3541, 3457, 3604, 3310, 3456, 3681, 3684, 3100, 3244, 3453, 3478, 3917, 3931, 3883, 4048, 3528, 3636, 3757, 3850, 4021, 4111, 4199, 4320, 4745, 4817

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

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

Row sums of triangle A268719.

Cf. A003188, A006068, A268715.

(define (A268720 n) (add (lambda (k) (A268715bi n k)) 0 n)) ;; Code for A268715bi given in A268715.
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

A268933 Permutation of nonnegative integers: a(0) = 0, for n >= 1, a(n) = A268717(1 + A268831(n-1)).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 12, 15, 13, 9, 11, 14, 10, 29, 31, 26, 30, 8, 24, 27, 25, 53, 55, 50, 54, 16, 48, 51, 49, 28, 20, 23, 21, 17, 19, 22, 18, 101, 103, 98, 102, 32, 96, 99, 97, 44, 36, 39, 37, 33, 35, 38, 34, 52, 60, 63, 61, 57, 59, 62, 58, 45, 47, 42, 46, 56, 40, 43, 41, 197, 199, 194, 198, 64, 192, 195, 193, 76

Offset: 0

Antti Karttunen, Feb 16 2016

nonn

The seventh "shifted power" of A268717.

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

Inverse: A268934.
Row 7 of A268820.
From term a(7) onward (4, 12, 15, 13, 9, 11, ...) also row 4 of A268715.
Cf. A003188, A006068, A268717, A268831.

(define (A268933 n) (if (zero? n) n (A268717 (+ 1 (A268831 (- n 1))))))

a(0) = 0, for n >= 1, a(n) = A268717(1 + A268831(n-1)).

cp's OEIS Frontend

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

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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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A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

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A268725 Square array A(i,j) = A003188(A006068(i) * A006068(j)), read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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

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A268714 Square array A(i,j) = A006068(i) + A006068(j), read by antidiagonals.

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