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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).

%I A286147 #20 Jun 02 2025 12:21:51
%S A286147 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,
%T A286147 49,84,35,43,52,62,73,85,98,112,44,34,25,51,10,72,59,97,144,54,64,33,
%U A286147 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
%N 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).
%C A286147 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 A286147 Antti Karttunen, <a href="/A286147/b286147.txt">Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array</a>
%H A286147 MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286147 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, ...].
%e A286147 The top left 0 .. 12 x 0 .. 12 corner of the array:
%e A286147     0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
%e A286147     4,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118
%e A286147    12,  18,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150
%e A286147    24,  17,  11,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186
%e A286147    40,  50,  61,  73,  10,  16,  23,  31, 148, 166, 185, 205,  86
%e A286147    60,  49,  85,  72,  22,  15,  39,  30, 184, 165, 225, 204, 114
%e A286147    84,  98,  59,  71,  38,  48,  21,  29, 224, 246, 183, 203, 146
%e A286147   112,  97,  83,  70,  58,  47,  37,  28, 268, 245, 223, 202, 182
%e A286147   144, 162, 181, 201, 222, 244, 267, 291,  36,  46,  57,  69,  82
%e A286147   180, 161, 221, 200, 266, 243, 315, 290,  56,  45,  81,  68, 110
%e A286147   220, 242, 179, 199, 314, 340, 265, 289,  80,  94,  55,  67, 142
%e A286147   264, 241, 219, 198, 366, 339, 313, 288, 108,  93,  79,  66, 178
%e A286147   312, 338, 365, 393, 218, 240, 263, 287, 140, 158, 177, 197,  78
%t A286147 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 *)
%o A286147 (Scheme)
%o A286147 (define (A286147 n) (A286147bi (A002262 n) (A025581 n)))
%o A286147 (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).
%o A286147 (Python)
%o A286147 def T(a, b): return ((a + b)**2 + 3*a + b)//2
%o A286147 def A(n, k): return T(n^k, k)
%o A286147 for n in range(21): print([A(n - k, k) for k in range(n + 1)]) # _Indranil Ghosh_, May 21 2017
%Y A286147 Transpose: A286145.
%Y A286147 Cf. A000096 (row 0), A046092 (column 0), A000217 (main diagonal).
%Y A286147 Cf. A003987, A001477, A286108, A286109, A286150, A286151.
%K A286147 nonn,tabl
%O A286147 0,2
%A A286147 _Antti Karttunen_, May 03 2017