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%I #10 Apr 02 2016 22:58:42
%S 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,
%T 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,
%U 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
%N 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.
%C Refer to A269845, but change to n+2 X n instead of n+1 X 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) Even terms of row sum is A002492. (ii) Odd terms of row sum/2 is A100157. See illustration in links.
%H Kival Ngaokrajang, <a href="/A270309/a270309.pdf">Illustration of initial terms</a>, <a href="/A270309/a270309_1.pdf">Row sum</a>
%e Irregular triangle begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
%e 1 1, 1
%e 2 1, 1, 1, 1
%e 3 9, 4, 1, 1, 4, 9
%e 4 4, 1, 1, 4, 4, 1, 1, 4
%e 5 25, 4, 9, 16, 1, 1, 16, 9, 4, 25
%e 6 9, 1, 4, 4, 1, 9, 9, 1, 4, 4, 1, 9
%e 7 49, 4, 25, 16, 9, 36, 1, 1, 36, 9, 16, 25, 4, 49
%e 8 16, 1, 9, 4, 4, 9, 1, 16, 16, 1, 9, 4, 4, 9, 1, 16
%e ...
%o (Small Basic)
%o For n=1 To 20
%o c=1
%o For k=1 To 2*n
%o If k<=n then
%o If Math.Remainder(n,2)=0 Then
%o If Math.remainder(k,2)=0 Then
%o t[n][k]=k/2
%o Else
%o t[n][k]=math.Floor(n/2-(k/2-1/2))
%o EndIf
%o Else
%o If Math.remainder(k,2)=0 Then
%o t[n][k]=k
%o Else
%o t[n][k]=(n-k)+1
%o EndIf
%o EndIf
%o TextWindow.Write(t[n][k]*t[n][k]+ ", ")
%o Else
%o t[n][k]=t[n][k-c]
%o TextWindow.write(t[n][k]*t[n][k]+ ", ")
%o c=c+2
%o EndIf
%o EndFor
%o EndFor
%Y Cf. A002492, A100157, A269845.
%K nonn,tabf
%O 1,7
%A _Kival Ngaokrajang_, Mar 15 2016
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