|
%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
|
