A286147 Square array read by antidiagonals: A(n,k) = T(n XOR k, n), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987)

0, 2, 4, 5, 1, 12, 9, 13, 18, 24, 14, 8, 3, 17, 40, 20, 26, 7, 11, 50, 60, 27, 19, 42, 6, 61, 49, 84, 35, 43, 52, 62, 73, 85, 98, 112, 44, 34, 25, 51, 10, 72, 59, 97, 144, 54, 64, 33, 41, 16, 22, 71, 83, 162, 180, 65, 53, 88, 32, 23, 15, 38, 70, 181, 161, 220, 77, 89, 102, 116, 31, 39, 48, 58, 201, 221, 242, 264, 90, 76, 63, 101, 148, 30, 21, 47, 222, 200, 179, 241, 312

Offset

0,2

Comments

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Links

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array

MathWorld, Pairing Function

Formula

A(n,k) = T(A003987(n,k), n), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

Example

The top left 0 .. 12 x 0 .. 12 corner of the array:
    0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
    4,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118
   12,  18,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150
   24,  17,  11,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186
   40,  50,  61,  73,  10,  16,  23,  31, 148, 166, 185, 205,  86
   60,  49,  85,  72,  22,  15,  39,  30, 184, 165, 225, 204, 114
   84,  98,  59,  71,  38,  48,  21,  29, 224, 246, 183, 203, 146
  112,  97,  83,  70,  58,  47,  37,  28, 268, 245, 223, 202, 182
  144, 162, 181, 201, 222, 244, 267, 291,  36,  46,  57,  69,  82
  180, 161, 221, 200, 266, 243, 315, 290,  56,  45,  81,  68, 110
  220, 242, 179, 199, 314, 340, 265, 289,  80,  94,  55,  67, 142
  264, 241, 219, 198, 366, 339, 313, 288, 108,  93,  79,  66, 178
  312, 338, 365, 393, 218, 240, 263, 287, 140, 158, 177, 197,  78

Mathematica

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], k]; Table[A[n - k, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

Prog

(Scheme)
(define (A286147 n) (A286147bi (A002262 n) (A025581 n)))
(define (A286147bi row col) (let ((a (A003987bi row col)) (b row)) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).
(Python)
def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k)
for n in range(21): print([A(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

Crossrefs

Transpose: A286145.

Cf. A000096 (row 0), A046092 (column 0), A000217 (main diagonal).

Cf. A003987, A001477, A286108, A286109, A286150, A286151.

Sequence in context: A274316 A075884 A030750 * A340755 A369772 A059215

Adjacent sequences: A286144 A286145 A286146 * A286148 A286149 A286150

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved