A286145 Square array read by antidiagonals: A(n,k) = T(n XOR k, k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

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

Offset: 0

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

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), ...

Examples

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

Links

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

Crossrefs

Transpose: A286147.

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

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

Programs

Mathematica

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

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(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

Scheme

(define (A286145 n) (A286145bi (A002262 n) (A025581 n)))
(define (A286145bi row col) (let ((a (A003987bi row col)) (b col)) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).

Formula

A(n,k) = T(A003987(n,k), k), 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, ...].