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A286109
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Square array read by antidiagonals: A(n,k) = T(n XOR k, 2*(n AND k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and XOR is bitwise-xor (A003987).
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7
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0, 2, 2, 5, 3, 5, 9, 9, 9, 9, 14, 12, 10, 12, 14, 20, 20, 16, 16, 20, 20, 27, 25, 27, 21, 27, 25, 27, 35, 35, 35, 35, 35, 35, 35, 35, 44, 42, 40, 42, 36, 42, 40, 42, 44, 54, 54, 50, 50, 46, 46, 50, 50, 54, 54, 65, 63, 65, 59, 57, 55, 57, 59, 65, 63, 65, 77, 77, 77, 77, 69, 69, 69, 69, 77, 77, 77, 77, 90, 88, 86, 88, 90, 80, 78, 80, 90, 88, 86, 88, 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), 2*A004198(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, 3, 9, 12, 20, 25, 35, 42, 54, 63, 77, 88, 104
5, 9, 10, 16, 27, 35, 40, 50, 65, 77, 86, 100, 119
9, 12, 16, 21, 35, 42, 50, 59, 77, 88, 100, 113, 135
14, 20, 27, 35, 36, 46, 57, 69, 90, 104, 119, 135, 144
20, 25, 35, 42, 46, 55, 69, 80, 104, 117, 135, 150, 162
27, 35, 40, 50, 57, 69, 78, 92, 119, 135, 148, 166, 181
35, 42, 50, 59, 69, 80, 92, 105, 135, 150, 166, 183, 201
44, 54, 65, 77, 90, 104, 119, 135, 136, 154, 173, 193, 214
54, 63, 77, 88, 104, 117, 135, 150, 154, 171, 193, 212, 236
65, 77, 86, 100, 119, 135, 148, 166, 173, 193, 210, 232, 259
77, 88, 100, 113, 135, 150, 166, 183, 193, 212, 232, 253, 283
90, 104, 119, 135, 144, 162, 181, 201, 214, 236, 259, 283, 300
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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], 2*BitAnd[n, k]]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 20 2017 *)
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PROG
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(Scheme)
(define (A286109bi row col) (let ((a (A003987bi row col)) (b (* 2 (A004198bi row col)))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003987bi and A004198bi implement bitwise-xor (A003987) and bitwise-and (A004198).
(Python)
def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, 2*(n&k))
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 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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