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A286145
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Square array read by antidiagonals: A(n,k) = T(n XOR k, 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, 4, 2, 12, 1, 5, 24, 18, 13, 9, 40, 17, 3, 8, 14, 60, 50, 11, 7, 26, 20, 84, 49, 61, 6, 42, 19, 27, 112, 98, 85, 73, 62, 52, 43, 35, 144, 97, 59, 72, 10, 51, 25, 34, 44, 180, 162, 83, 71, 22, 16, 41, 33, 64, 54, 220, 161, 181, 70, 38, 15, 23, 32, 88, 53, 65, 264, 242, 221, 201, 58, 48, 39, 31, 116, 102, 89, 77, 312, 241, 179, 200, 222, 47, 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), 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, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312
2, 1, 18, 17, 50, 49, 98, 97, 162, 161, 242, 241, 338
5, 13, 3, 11, 61, 85, 59, 83, 181, 221, 179, 219, 365
9, 8, 7, 6, 73, 72, 71, 70, 201, 200, 199, 198, 393
14, 26, 42, 62, 10, 22, 38, 58, 222, 266, 314, 366, 218
20, 19, 52, 51, 16, 15, 48, 47, 244, 243, 340, 339, 240
27, 43, 25, 41, 23, 39, 21, 37, 267, 315, 265, 313, 263
35, 34, 33, 32, 31, 30, 29, 28, 291, 290, 289, 288, 287
44, 64, 88, 116, 148, 184, 224, 268, 36, 56, 80, 108, 140
54, 53, 102, 101, 166, 165, 246, 245, 46, 45, 94, 93, 158
65, 89, 63, 87, 185, 225, 183, 223, 57, 81, 55, 79, 177
77, 76, 75, 74, 205, 204, 203, 202, 69, 68, 67, 66, 197
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], 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 (A286145bi row col) (let ((a (A003987bi row col)) (b col)) (/ (+ (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, 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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