A268317 -id:A268317 - cp's OEIS frontend

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

Kival Ngaokrajang, Mar 06 2016

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.

			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
...

Kival Ngaokrajang, Illustration of initial terms, Row sum

Cf. A100157, A258582, A268317.

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 *)

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, ", ")))

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.

A268318 Irregular triangle read by rows: T(n,k) gives the row sums in the table Fib(n+1) X Fib(n), where k = 1..Fib(n+1), and 1's are assigned to cells on the longest diagonal path.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2

Offset: 0

Kival Ngaokrajang, Feb 01 2016

Inspired by sun flower spirals which come in  Fib(i) and Fib(i+1) numbers in opposite directions. The present Fib(n+1) X Fib(n) table has the following properties:
(i) Columns sum create the irregular triangle A268317.
(ii) Rows sum create the present irregular triangle.
(iii) The row sums of each of these irregular triangles is conjectured to be A000071.
(iv) The first differences of the sequence of half of the  voids (0's) are conjectured to give A191797.
See illustrations in the links of A268317.

			Irregular triangle begins:
0
1
1 1
1 2 1
1 2 1 2 1
1 2 1 2 2 1 2 1
1 2 1 2 2 1 2 1 2 2 1 2 1
1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1
...

Cf. A000071, A191797, A268317.

cp's OEIS Frontend

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.

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