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A082233
Square array T(n,k) = 2*n + k, read by antidiagonals in a zigzag fashion, n >= 0 and k >= 1.
3
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
OFFSET
0,2
COMMENTS
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
LINKS
EXAMPLE
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.
MATHEMATICA
Flatten@Table[If[EvenQ[n], #, Reverse[#]] &[Range[n, 2 n - 1]], {n, 11}] (* Ivan Neretin, Aug 24 2017 *)
CROSSREFS
Cf. A000326, A016777, A082234, A229035 (partial sums).
Sequence in context: A330080 A068508 A137403 * A330806 A058981 A117339
KEYWORD
easy,nonn,tabl
AUTHOR
Amarnath Murthy, Apr 10 2003
EXTENSIONS
More terms from Michel Marcus, Jan 20 2013
New definition from Joerg Arndt and Michel Marcus, Jan 20 2013, corrected R. J. Mathar, Sep 21 2013
STATUS
approved