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A286150
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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).
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6
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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, 27, 35, 43, 52, 62, 62, 52, 43, 35, 44, 34, 25, 51, 10, 51, 25, 34, 44, 54, 64, 33, 41, 16, 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
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OFFSET
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0,2
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COMMENTS
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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), ...
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LINKS
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FORMULA
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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, ...].
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EXAMPLE
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The top left 0 .. 12 x 0 .. 12 corner of the array:
0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90
2, 1, 13, 8, 26, 19, 43, 34, 64, 53, 89, 76, 118
5, 13, 3, 7, 42, 52, 25, 33, 88, 102, 63, 75, 150
9, 8, 7, 6, 62, 51, 41, 32, 116, 101, 87, 74, 186
14, 26, 42, 62, 10, 16, 23, 31, 148, 166, 185, 205, 86
20, 19, 52, 51, 16, 15, 39, 30, 184, 165, 225, 204, 114
27, 43, 25, 41, 23, 39, 21, 29, 224, 246, 183, 203, 146
35, 34, 33, 32, 31, 30, 29, 28, 268, 245, 223, 202, 182
44, 64, 88, 116, 148, 184, 224, 268, 36, 46, 57, 69, 82
54, 53, 102, 101, 166, 165, 246, 245, 46, 45, 81, 68, 110
65, 89, 63, 87, 185, 225, 183, 223, 57, 81, 55, 67, 142
77, 76, 75, 74, 205, 204, 203, 202, 69, 68, 67, 66, 178
90, 118, 150, 186, 86, 114, 146, 182, 82, 110, 142, 178, 78
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MATHEMATICA
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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 *)
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PROG
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(Scheme)
(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).
(Python)
def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, min(n, k))
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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