A269845 Irregular triangle read by rows: T(n,k) = (k/2+1/2)^2 if odd-k otherwise T(n,k) = (n-k/2)^2 where n >= 1, k = 0..2*n-1.

1, 1, 4, 1, 1, 4, 9, 1, 4, 4, 1, 9, 16, 1, 9, 4, 4, 9, 1, 16, 25, 1, 16, 4, 9, 9, 4, 16, 1, 25, 36, 1, 25, 4, 16, 9, 9, 16, 4, 25, 1, 36, 49, 1, 36, 4, 25, 9, 16, 16, 9, 25, 4, 36, 1, 49, 64, 1, 49, 4, 36, 9, 25, 16, 16, 25, 9, 36, 4, 49, 1, 64, 81, 1, 64, 4, 49, 9, 36, 16, 25, 25, 16, 36, 9, 49, 4, 64, 1, 81, 100, 1, 81, 4, 64, 9, 49, 16, 36, 25, 25, 36, 16, 49

Offset

1,3

Comments

Inspired by A268317, but change to n+1 X n instead of Fib(n+1) X Fib(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) Odd terms of row sum/2 is A100157. (ii) Even terms of row sum/2 is A258582. See illustration in links.

Links

Table of n, a(n) for n=1..104.

Kival Ngaokrajang, Illustration of initial terms, Row sum

Formula

T(n,k) = (k/2+1/2)^2 if odd-k, T(n,k) = (n-k/2)^2 if even-k; n >= 1, k = 0..2*n-1.

Example

Irregular triangle begins:
n\k 0  1   2  3   4  5   6   7   8   9  10 11 12  13 14  15 ...
1   1, 1
2   4, 1,  1, 4
3   9, 1,  4, 4,  1, 9
4  16, 1,  9, 4,  4, 9,  1, 16
5  25, 1, 16, 4,  9, 9,  4, 16,  1, 25
6  36, 1, 25, 4, 16, 9,  9, 16,  4, 25, 1, 36
7  49, 1, 36, 4, 25, 9, 16, 16,  9, 25, 4, 36, 1, 49
8  64, 1, 49, 4, 36, 9, 25, 16, 16, 25, 9, 36, 4, 49, 1, 64
...

Mathematica

Table[If[OddQ@ k, (k/2 + 1/2)^2, (n - k/2)^2], {n, 8}, {k, 0, 2 n - 1}] // Flatten (* Michael De Vlieger, Apr 01 2016 *)

Prog

(PARI) for (n = 1, 20, for (k = 0, 2*n-1, if (Mod(k, 2)==0, t = (n-k/2)^2, t = (k/2+1/2)^2); print1(t, ", ")))

Crossrefs

Cf. A100157, A258582, A268317.

Sequence in context: A080061 A246595 A209571 * A124258 A001638 A133826

Adjacent sequences: A269842 A269843 A269844 * A269846 A269847 A269848

Keyword

nonn,tabf

Author

Kival Ngaokrajang, Mar 06 2016

Status

approved