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%I #23 Apr 10 2021 16:38:14
%S 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,
%T 27,28,37,47,58,62,52,43,35,36,29,23,48,14,51,25,34,44,45,56,30,39,19,
%U 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
%N 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).
%C 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), ...
%H Antti Karttunen, <a href="/A286151/b286151.txt">Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F 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).
%e The top left 0 .. 12 x 0 .. 12 corner of the array:
%e 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78
%e 2, 2, 11, 7, 22, 16, 37, 29, 56, 46, 79, 67, 106
%e 5, 13, 5, 8, 38, 47, 23, 30, 80, 93, 57, 68, 138
%e 9, 8, 7, 9, 58, 48, 39, 31, 108, 94, 81, 69, 174
%e 14, 26, 42, 62, 14, 19, 25, 32, 140, 157, 175, 194, 82
%e 20, 19, 52, 51, 16, 20, 41, 33, 176, 158, 215, 195, 110
%e 27, 43, 25, 41, 23, 39, 27, 34, 216, 237, 177, 196, 142
%e 35, 34, 33, 32, 31, 30, 29, 35, 260, 238, 217, 197, 178
%e 44, 64, 88, 116, 148, 184, 224, 268, 44, 53, 63, 74, 86
%e 54, 53, 102, 101, 166, 165, 246, 245, 46, 54, 87, 75, 114
%e 65, 89, 63, 87, 185, 225, 183, 223, 57, 81, 65, 76, 146
%e 77, 76, 75, 74, 205, 204, 203, 202, 69, 68, 67, 77, 182
%e 90, 118, 150, 186, 86, 114, 146, 182, 82, 110, 142, 178, 90
%t 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 *)
%o (Scheme)
%o (define (A286151 n) (A286151bi (A002262 n) (A025581 n)))
%o (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).
%o (Python)
%o def T(a, b): return ((a + b)**2 + 3*a + b)//2
%o def A(n, k): return T(n^k, k) if n>k else T(n, n^k)
%o for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # _Indranil Ghosh_, May 20 2017
%Y Cf. A000217 (row 0), A000096 (column 0 and the main diagonal).
%Y Cf. A001477, A003987, A286108, A286109, A286145, A286147, A286150.
%Y Cf. A286153 (same array without row 0 and column 0).
%K nonn,tabl
%O 0,3
%A _Antti Karttunen_, May 03 2017
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