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A082233
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Square array T(n,k) = 2*n + k, read by antidiagonals in a zigzag fashion, n >= 0 and k >= 1.
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3
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1, 2, 3, 5, 4, 3, 4, 5, 6, 7, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 13, 12, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 15, 14, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11
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OFFSET
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0,2
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COMMENTS
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The n-th row contains natural numbers starting from 2n+1. The 2n-th column contains even numbers beginning with 2n. The (2n-1)-th column contains odd numbers beginning with 2n-1. The n-th antidiagonal sum is given by pentagonal number A000326(n+1). The main diagonal is given by A016777.
For n >= 0 and k >= 1, the term T(n,k) occupies position m = (n+k)*(n+k-1)/2 + k*(1 - (-1)^(n+k))/2 + (n+1)*(1 + (-1)^(n+k))/2 - 1 in the sequence (a(s): s >= 0), i.e., a(m) = T(n,k). - Petros Hadjicostas, Feb 26 2021
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LINKS
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EXAMPLE
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In the following square array (T(n,k): n >= 0, k >= 1), numbers are entered like this: T(0,1), T(0,2), T(1,1), T(2,1), T(1,2), T(0,3), T(0,4), T(1,3), T(2,2), T(3,1), T(4,1), T(3,2), ..., such that every entry is the arithmetic mean of the two diametrically opposite neighbors (wherever such a pair exists).
1 2 3 4 5 6 7 ...
3 4 5 6 7 8 9 ...
5 6 7 8 9 10 11 ...
7 8 9 10 11 12 13 ...
9 10 11 12 13 14 15 ...
...
The sequence (a(n): n >= 0) contains the numbers in the order in which they are entered in the above square array T.
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MATHEMATICA
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Flatten@Table[If[EvenQ[n], #, Reverse[#]] &[Range[n, 2 n - 1]], {n, 11}] (* Ivan Neretin, Aug 24 2017 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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