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
Ivan Neretin, Table of n, a(n) for n = 0..5049
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
KEYWORD
AUTHOR
Amarnath Murthy, Apr 10 2003
EXTENSIONS
More terms from Michel Marcus, Jan 20 2013
STATUS
approved