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%I #23 Mar 22 2017 10:31:52
%S 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,
%T 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,
%U 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
%N 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.
%C Inspired by A268317, but change to n+1 X n instead of Fib(n+1) X Fib(n).
%C 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.
%H Kival Ngaokrajang, <a href="/A269845/a269845_1.pdf">Illustration of initial terms</a>, <a href="/A269845/a269845_2.pdf">Row sum</a>
%F 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.
%e Irregular triangle begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e 1 1, 1
%e 2 4, 1, 1, 4
%e 3 9, 1, 4, 4, 1, 9
%e 4 16, 1, 9, 4, 4, 9, 1, 16
%e 5 25, 1, 16, 4, 9, 9, 4, 16, 1, 25
%e 6 36, 1, 25, 4, 16, 9, 9, 16, 4, 25, 1, 36
%e 7 49, 1, 36, 4, 25, 9, 16, 16, 9, 25, 4, 36, 1, 49
%e 8 64, 1, 49, 4, 36, 9, 25, 16, 16, 25, 9, 36, 4, 49, 1, 64
%e ...
%t 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 *)
%o (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, ", ")))
%Y Cf. A100157, A258582, A268317.
%K nonn,tabf
%O 1,3
%A _Kival Ngaokrajang_, Mar 06 2016
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