cp's OEIS Frontend

A270309 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.

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