A268714 -id:A268714 - cp's OEIS frontend

A268716 a(n) = 2*A006068(n); main diagonal of A268714.

0, 2, 6, 4, 14, 12, 8, 10, 30, 28, 24, 26, 16, 18, 22, 20, 62, 60, 56, 58, 48, 50, 54, 52, 32, 34, 38, 36, 46, 44, 40, 42, 126, 124, 120, 122, 112, 114, 118, 116, 96, 98, 102, 100, 110, 108, 104, 106, 64, 66, 70, 68, 78, 76, 72, 74, 94, 92, 88, 90, 80, 82, 86, 84, 254, 252, 248, 250, 240, 242, 246, 244, 224, 226

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

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

Cf. A001969, A006068.
Main diagonal of array A268714.
Row 3 and column 3 of array A268724.

def A268716(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k<<1 # Chai Wah Wu, Jun 30 2022

Scheme
```
(define (A268716 n) (* 2 (A006068 n)))
```

a(n) = 2*A006068(n).
a(n) = A006068(A001969(n+1)).
a(n) = A268714(n,n).

A268836 Antidiagonal sums of array A268714: a(n) = Sum_{k=0..n} A006068(n)+A006068(n-k).

Original entry on oeis.org

0, 2, 8, 12, 26, 38, 46, 56, 86, 114, 138, 164, 180, 198, 220, 240, 302, 362, 418, 476, 524, 574, 628, 680, 712, 746, 784, 820, 866, 910, 950, 992, 1118, 1242, 1362, 1484, 1596, 1710, 1828, 1944, 2040, 2138, 2240, 2340, 2450, 2558, 2662, 2768, 2832, 2898, 2968, 3036, 3114, 3190, 3262, 3336, 3430, 3522, 3610, 3700

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

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

Cf. A006068, A268714.
Cf. also A268837, A268721.
Partial sums of A268716.

(define (A268836 n) (add (lambda (k) (+ (A006068 k) (A006068 (- (+ n 0) 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} A006068(n)+A006068(n-k).

A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by 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, 1, 2, 3, 2, 3, 6, 6, 3, 4, 2, 5, 2, 4, 5, 12, 7, 7, 12, 5, 6, 4, 15, 6, 15, 4, 6, 7, 7, 13, 13, 13, 13, 7, 7, 8, 5, 4, 12, 9, 12, 4, 5, 8, 9, 24, 12, 5, 11, 11, 5, 12, 24, 9, 10, 8, 27, 4, 14, 10, 14, 4, 27, 8, 10, 11, 11, 25, 25, 10, 15, 15, 10, 25, 25, 11, 11, 12, 9, 8, 24, 29, 14, 12, 14, 29, 24, 8, 9, 12, 13, 13, 24, 9, 31, 31, 13, 13, 31, 31, 9, 24, 13, 13

Offset: 0

Antti Karttunen, Feb 12 2016

Each row n is row A006068(n) of array A268820 without its A006068(n) initial terms.

			The top left [0 .. 15] x [0 .. 15] section of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15
   1,  3,  6,  2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14
   2,  6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11
   3,  2,  7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10
   4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8, 24, 27, 25
   5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8, 25, 24
   6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
   7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24,  8
   8, 24, 27, 25, 29, 31, 26, 30, 17, 19, 22, 18, 28, 20, 23, 21
   9,  8, 25, 24, 31, 30, 27, 26, 19, 18, 23, 22, 29, 28, 21, 20
  10, 11,  8,  9, 26, 27, 24, 25, 22, 23, 20, 21, 30, 31, 28, 29
  11,  9, 24,  8, 30, 26, 25, 27, 18, 22, 21, 23, 31, 29, 20, 28
  12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31, 24, 25, 26, 27
  13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31, 29, 25, 27, 30, 26
  14, 10,  9, 11, 27, 25,  8, 24, 23, 21, 28, 20, 26, 30, 29, 31
  15, 14, 11, 10, 25, 24,  9,  8, 21, 20, 29, 28, 27, 26, 31, 30

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

Cf. A003188, A006068, A268714, A268820.
Main diagonal: A001969.
Row 0, column 0: A001477.
Row 1, column 1: A268717.
Antidiagonal sums: A268837.
Cf. A268719 (the lower triangular section).
Cf. also A268725.

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := BitXor @@ Table[Floor[ n/2^m], {m, 0, Log[2, n]}]; A006068[0]=0; A[i_, j_] := A003188[A006068[i] + A006068[j]]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

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, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268715 n) (A268715bi (A002262 n) (A025581 n)))
(define (A268715bi row col) (A003188 (+ (A006068 row) (A006068 col))))
;; Alternatively, extracting data from array A268820:
(define (A268715bi row col) (A268820bi (A006068 row) (+ (A006068 row) col)))

A(i,j) = A003188(A006068(i) + A006068(j)) = A003188(A268714(i,j)).

A(row,col) = A268820(A006068(row), (A006068(row)+col)).

A268724 Square array A(i,j) = 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, 3, 3, 2, 9, 2, 7, 6, 6, 7, 6, 21, 4, 21, 6, 4, 18, 14, 14, 18, 4, 5, 12, 12, 49, 12, 12, 5, 15, 15, 8, 42, 42, 8, 15, 15, 14, 45, 10, 28, 36, 28, 10, 45, 14, 12, 42, 30, 35, 24, 24, 35, 30, 42, 12, 13, 36, 28, 105, 30, 16, 30, 105, 28, 36, 13, 8, 39, 24, 98, 90, 20, 20, 90, 98, 24, 39, 8, 9, 24, 26, 84, 84, 60, 25, 60, 84, 84, 26, 24, 9

Offset: 1

Antti Karttunen, Feb 13 2016

			The top left [1 .. 15] x [1 .. 15] section of the array:
   1,  3,  2,  7,   6,  4,  5,  15,  14,  12,  13,   8,   9,  11,  10
   3,  9,  6,  21, 18, 12, 15,  45,  42,  36,  39,  24,  27,  33,  30
   2,  6,  4,  14, 12,  8, 10,  30,  28,  24,  26,  16,  18,  22,  20
   7, 21, 14,  49, 42, 28, 35, 105,  98,  84,  91,  56,  63,  77,  70
   6, 18, 12,  42, 36, 24, 30,  90,  84,  72,  78,  48,  54,  66,  60
   4, 12,  8,  28, 24, 16, 20,  60,  56,  48,  52,  32,  36,  44,  40
   5, 15, 10,  35, 30, 20, 25,  75,  70,  60,  65,  40,  45,  55,  50
  15, 45, 30, 105, 90, 60, 75, 225, 210, 180, 195, 120, 135, 165, 150
  14, 42, 28,  98, 84, 56, 70, 210, 196, 168, 182, 112, 126, 154, 140
  12, 36, 24,  84, 72, 48, 60, 180, 168, 144, 156,  96, 108, 132, 120
  13, 39, 26,  91, 78, 52, 65, 195, 182, 156, 169, 104, 117, 143, 130
   8, 24, 16,  56, 48, 32, 40, 120, 112,  96, 104,  64,  72,  88,  80
   9, 27, 18,  63, 54, 36, 45, 135, 126, 108, 117,  72,  81,  99,  90
  11, 33, 22,  77, 66, 44, 55, 165, 154, 132, 143,  88,  99, 121, 110
  10, 30, 20,  70, 60, 40, 50, 150, 140, 120, 130,  80,  90, 110, 100

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

Cf. A268725.
Cf. A006068 (row 1, column 1).
Cf. A268716 (row 3, column 3).
Cf. A268721 (the antidiagonal sums).
Cf. also A268714.

(define (A268724 n) (A268724bi (A002260 n) (A004736 n)))
(define (A268724bi row col) (* (A006068 row) (A006068 col)))

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

A(i,j) = A006068(A268725(i,j)).

cp's OEIS Frontend

A268716 a(n) = 2*A006068(n); main diagonal of A268714.

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A268836 Antidiagonal sums of array A268714: a(n) = Sum_{k=0..n} A006068(n)+A006068(n-k).

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A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A268724 Square array A(i,j) = 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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