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A285722
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Square array A(n,k) read by antidiagonals, A(n,n) = 0, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.
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9
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0, 1, 1, 2, 0, 3, 4, 3, 2, 6, 7, 5, 0, 5, 10, 11, 8, 6, 4, 9, 15, 16, 12, 9, 0, 8, 14, 21, 22, 17, 13, 10, 7, 13, 20, 28, 29, 23, 18, 14, 0, 12, 19, 27, 36, 37, 30, 24, 19, 15, 11, 18, 26, 35, 45, 46, 38, 31, 25, 20, 0, 17, 25, 34, 44, 55, 56, 47, 39, 32, 26, 21, 16, 24, 33, 43, 54, 66, 67, 57, 48, 40, 33, 27, 0, 23, 32, 42, 53, 65, 78, 79, 68, 58, 49, 41, 34, 28, 22, 31, 41, 52, 64, 77, 91
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OFFSET
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1,4
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COMMENTS
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The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
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LINKS
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FORMULA
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If n = k, A(n,k) = 0, if n > k, A(n,k) = T(n-k,k), otherwise [when n < k], A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N X N to N.
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EXAMPLE
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The top left 14 X 14 corner of the array:
0, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79
1, 0, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80
3, 2, 0, 6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81
6, 5, 4, 0, 10, 14, 19, 25, 32, 40, 49, 59, 70, 82
10, 9, 8, 7, 0, 15, 20, 26, 33, 41, 50, 60, 71, 83
15, 14, 13, 12, 11, 0, 21, 27, 34, 42, 51, 61, 72, 84
21, 20, 19, 18, 17, 16, 0, 28, 35, 43, 52, 62, 73, 85
28, 27, 26, 25, 24, 23, 22, 0, 36, 44, 53, 63, 74, 86
36, 35, 34, 33, 32, 31, 30, 29, 0, 45, 54, 64, 75, 87
45, 44, 43, 42, 41, 40, 39, 38, 37, 0, 55, 65, 76, 88
55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 0, 66, 77, 89
66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 0, 78, 90
78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 0, 91
91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 0
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MATHEMATICA
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A[n_, n_] = 0;
A[n_, k_] /; k == n-1 := (k^2 - k + 2)/2;
A[1, k_] := (k^2 - 3k + 4)/2;
A[n_, k_] /; 1 <= k <= n-2 := A[n, k] = A[n, k+1] + 1;
A[n_, k_] /; k > n := A[n, k] = A[n-1, k] + 1;
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PROG
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(Scheme)
(define (A285722bi row col) (cond ((= row col) 0) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))
(Python)
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return 0 if n == k else T(n - k, k) if n>k else T(n, k - n)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 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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