A269845 -id:A269845 - cp's OEIS frontend

A270309 Irregular triangle read by rows: T(n,k) = ((n-k)+1)^2 if odd-n and odd-k; T(n,k) = k^2 if odd-n and even-k; T(n,k) = (n/2-(k/2-1/2))^2 if even-n and odd-k; T(n,k) = (k/2+1)^2 if even-n and even-k; where n >= 1, k = 1..2*n.

1, 1, 1, 1, 1, 1, 9, 4, 1, 1, 4, 9, 4, 1, 1, 4, 4, 1, 1, 4, 25, 4, 9, 16, 1, 1, 16, 9, 4, 25, 9, 1, 4, 4, 1, 9, 9, 1, 4, 4, 1, 9, 49, 4, 25, 16, 9, 36, 1, 1, 36, 9, 16, 25, 4, 49, 16, 1, 9, 4, 4, 9, 1, 16, 16, 1, 9, 4, 4, 9, 1, 16, 81, 4, 49, 16, 25, 36, 9, 64, 1, 1, 64, 9, 36, 25, 16, 49, 4, 81, 25, 1, 16, 4, 9, 9, 4, 16, 1, 25, 25, 1, 16, 4, 9, 9, 4, 16, 1, 25

Offset: 1

Kival Ngaokrajang, Mar 15 2016

Refer to A269845, but change to n+2 X n instead of n+1 X n.

There are triangles appearing along main diagonal. If the area of the smallest triangles are defined as 1, then the areas of all other triangles seem to be square numbers. Conjectures: (i) Even terms of row sum is A002492. (ii) Odd terms of row sum/2 is A100157. See illustration in links.

			Irregular triangle begins:
n\k  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  ...
1    1,  1
2    1,  1,  1,  1
3    9,  4,  1,  1,  4,  9
4    4,  1,  1,  4,  4,  1,  1,  4
5   25,  4,  9, 16,  1,  1, 16,  9,  4, 25
6    9,  1,  4,  4,  1,  9,  9,  1,  4,  4,  1,  9
7   49,  4, 25, 16,  9, 36,  1,  1, 36,  9, 16, 25,  4, 49
8   16,  1,  9,  4,  4,  9,  1, 16, 16,  1,  9,  4,  4,  9,  1, 16
...

Kival Ngaokrajang, Illustration of initial terms, Row sum

Cf. A002492, A100157, A269845.

cp's OEIS Frontend

A270309 Irregular triangle read by rows: T(n,k) = ((n-k)+1)^2 if odd-n and odd-k; T(n,k) = k^2 if odd-n and even-k; T(n,k) = (n/2-(k/2-1/2))^2 if even-n and odd-k; T(n,k) = (k/2+1)^2 if even-n and even-k; where n >= 1, k = 1..2*n.

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