A286151 Square array read by descending antidiagonals: If n > k, A(n,k) = T(n XOR k, k), and otherwise A(n,k) = T(n, n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

0, 1, 2, 3, 2, 5, 6, 11, 13, 9, 10, 7, 5, 8, 14, 15, 22, 8, 7, 26, 20, 21, 16, 38, 9, 42, 19, 27, 28, 37, 47, 58, 62, 52, 43, 35, 36, 29, 23, 48, 14, 51, 25, 34, 44, 45, 56, 30, 39, 19, 16, 41, 33, 64, 54, 55, 46, 80, 31, 25, 20, 23, 32, 88, 53, 65, 66, 79, 93, 108, 32, 41, 39, 31, 116, 102, 89, 77, 78, 67, 57, 94, 140, 33, 27, 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,   1,   3,   6,  10,  15,  21,  28,  36,  45,  55,  66,  78
   2,   2,  11,   7,  22,  16,  37,  29,  56,  46,  79,  67, 106
   5,  13,   5,   8,  38,  47,  23,  30,  80,  93,  57,  68, 138
   9,   8,   7,   9,  58,  48,  39,  31, 108,  94,  81,  69, 174
  14,  26,  42,  62,  14,  19,  25,  32, 140, 157, 175, 194,  82
  20,  19,  52,  51,  16,  20,  41,  33, 176, 158, 215, 195, 110
  27,  43,  25,  41,  23,  39,  27,  34, 216, 237, 177, 196, 142
  35,  34,  33,  32,  31,  30,  29,  35, 260, 238, 217, 197, 178
  44,  64,  88, 116, 148, 184, 224, 268,  44,  53,  63,  74,  86
  54,  53, 102, 101, 166, 165, 246, 245,  46,  54,  87,  75, 114
  65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  65,  76, 146
  77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  77, 182
  90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  90
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
• Eric Weisstein's World of Mathematics, Pairing Function

Crossrefs

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

Cf. A001477, A003987, A286108, A286109, A286145, A286147, A286150.

Cf. A286153 (same array without row 0 and column 0).

Programs

Mathematica

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

Python

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k) if n>k else T(n, n^k)
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 2017

Scheme

(define (A286151 n) (A286151bi (A002262 n) (A025581 n)))
(define (A286151bi row col) (define (pairA001477bi a b) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2)) (cond ((> row col) (pairA001477bi (A003987bi row col) col)) (else (pairA001477bi row (A003987bi col row))))) ;; Where A003987bi implements bitwise-xor (A003987).

Formula

If n > k, A(n,k) = T(A003987(n,k),k), otherwise A(n,k) = T(n,A003987(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).