A268724 -id:A268724 - cp's OEIS frontend

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), ...

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)).

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

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

Original entry on oeis.org

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).

A268721 Convolution of A006068 (inverse of Gray code) with itself: a(n) = Sum_{k=1..n+1} A006068(k) * A006068(1+n-k).

Original entry on oeis.org

0, 1, 6, 13, 26, 58, 72, 107, 160, 230, 286, 440, 558, 599, 696, 851, 1032, 1298, 1510, 1826, 2122, 2353, 2624, 3294, 3884, 4335, 4870, 5001, 5242, 5722, 6048, 6699, 7424, 8226, 8990, 10166, 11226, 12069, 13048, 14384, 15664, 16885, 18134, 19071, 20094, 21276, 22360, 25150, 27788, 30091, 32582, 34343, 36262

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1023

Antidiagonal sums of array A268724.

Cf. A006068

(define A268721 (CONVOLVE 1 A006068 A006068)) ;; This version requires Antti Karttunen's IntSeq-library.
;; More stand-alone version:
(define (A268721 n) (add (lambda (k) (* (A006068 k) (A006068 (- (+ n 1) k)))) 1 (+ n 1)))
(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=1..n+1} A006068(k) * A006068(1+n-k).

cp's OEIS Frontend

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

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

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A268721 Convolution of A006068 (inverse of Gray code) with itself: a(n) = Sum_{k=1..n+1} A006068(k) * A006068(1+n-k).

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