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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.

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