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

0,3

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

Eric Weisstein's World of Mathematics, Pairing Function

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).

Example

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

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 *)

Prog

(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).
(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

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).

Sequence in context: A182880 A182898 A133684 * A192138 A175264 A025473

Adjacent sequences: A286148 A286149 A286150 * A286152 A286153 A286154

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved