login
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.
1
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, 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, 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
OFFSET
1,7
COMMENTS
Refer to A269845, but change to n+2 X n instead of n+1 X 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) Even terms of row sum is A002492. (ii) Odd terms of row sum/2 is A100157. See illustration in links.
EXAMPLE
Irregular triangle begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
1 1, 1
2 1, 1, 1, 1
3 9, 4, 1, 1, 4, 9
4 4, 1, 1, 4, 4, 1, 1, 4
5 25, 4, 9, 16, 1, 1, 16, 9, 4, 25
6 9, 1, 4, 4, 1, 9, 9, 1, 4, 4, 1, 9
7 49, 4, 25, 16, 9, 36, 1, 1, 36, 9, 16, 25, 4, 49
8 16, 1, 9, 4, 4, 9, 1, 16, 16, 1, 9, 4, 4, 9, 1, 16
...
PROG
(Small Basic)
For n=1 To 20
c=1
For k=1 To 2*n
If k<=n then
If Math.Remainder(n, 2)=0 Then
If Math.remainder(k, 2)=0 Then
t[n][k]=k/2
Else
t[n][k]=math.Floor(n/2-(k/2-1/2))
EndIf
Else
If Math.remainder(k, 2)=0 Then
t[n][k]=k
Else
t[n][k]=(n-k)+1
EndIf
EndIf
TextWindow.Write(t[n][k]*t[n][k]+ ", ")
Else
t[n][k]=t[n][k-c]
TextWindow.write(t[n][k]*t[n][k]+ ", ")
c=c+2
EndIf
EndFor
EndFor
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Kival Ngaokrajang, Mar 15 2016
STATUS
approved