A286150 - OEIS

%I #21 Apr 30 2021 17:14:18

%S 0,2,2,5,1,5,9,13,13,9,14,8,3,8,14,20,26,7,7,26,20,27,19,42,6,42,19,

%T 27,35,43,52,62,62,52,43,35,44,34,25,51,10,51,25,34,44,54,64,33,41,16,

%U 16,41,33,64,54,65,53,88,32,23,15,23,32,88,53,65,77,89,102,116,31,39,39,31,116,102,89,77,90,76,63,101,148,30,21,30,148,101,63,76,90

%N Square array read by antidiagonals: A(n,k) = T(n XOR k, min(n,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="/A286150/b286150.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="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%F A(n,k) = T(A003987(n,k), min(n,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, ...].

%e The top left 0 .. 12 x 0 .. 12 corner of the array:

%e    0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90

%e    2,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118

%e    5,  13,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150

%e    9,   8,   7,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186

%e   14,  26,  42,  62,  10,  16,  23,  31, 148, 166, 185, 205,  86

%e   20,  19,  52,  51,  16,  15,  39,  30, 184, 165, 225, 204, 114

%e   27,  43,  25,  41,  23,  39,  21,  29, 224, 246, 183, 203, 146

%e   35,  34,  33,  32,  31,  30,  29,  28, 268, 245, 223, 202, 182

%e   44,  64,  88, 116, 148, 184, 224, 268,  36,  46,  57,  69,  82

%e   54,  53, 102, 101, 166, 165, 246, 245,  46,  45,  81,  68, 110

%e   65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  55,  67, 142

%e   77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  66, 178

%e   90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  78

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

%o (Scheme)

%o (define (A286150 n) (A286150bi (A002262 n) (A025581 n)))

%o (define (A286150bi row col) (let ((a (A003987bi row col)) (b (min col row))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; 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, min(n, k))

%o for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # _Indranil Ghosh_, May 21 2017

%Y Cf. A000096 (row 0 & column 0), A000217 (main diagonal).

%Y Cf. A003987, A001477, A286108, A286109, A286145, A286147, A286151.

%K nonn,tabl

%O 0,2

%A _Antti Karttunen_, May 03 2017