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, 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.

Links

Table of n, a(n) for n=1..110.

Kival Ngaokrajang, Illustration of initial terms, Row sum

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

Cf. A002492, A100157, A269845.

Sequence in context: A056897 A263192 A355282 * A010158 A336274 A286229

Adjacent sequences: A270306 A270307 A270308 * A270310 A270311 A270312

Keyword

nonn,tabf

Author

Kival Ngaokrajang, Mar 15 2016

Status

approved