For the average to be n^2, the sum of the n squares must be n^3, so that's what we seek.

When n is 2161, the sum of the squares from the 989th (978121) through the 3149th (9916201) square is 10091699281, the 2161st cube; when divided by 2161 to get the average that average is 2161^2.

The program output:

47 22 484 68 4624 103823

2161 989 978121 3149 9916201 10091699281

After running the program, entering 47,2161 into Sloane's OEIS finds A189173, Integers m such that m^3 is the sum of squares of m consecutive integers. The list is preceded by zero and 1 and followed by 99359, 4568353, 210044879, 9657496081, 444034774847, 20415942146881, 938689303981679, 43159292041010353, 1984388744582494559, 91238722958753739361, 4194996867358089516047, 192878617175513363998801, 8868221393206256654428799, 407745305470312292739725953, 18747415830241159209372965039. The recursive formula is a(n) = 46*a(n-1) - a(n-2) where the a values are the n values of the puzzle.

The program:

DefDbl A-Z

Dim squares(4000), sumsquares(4000), cubes(4000), crlf$

Private Sub Form_Load()

 Text1.Text = ""

 crlf$ = Chr(13) + Chr(10)

 Form1.Visible = True

 

  For i = 1 To 4000

    squares(i) = i * i

    sumsquares(i) = sumsquares(i - 1) + squares(i)

    cubes(i) = squares(i) * i

  Next

  

  For n = 2 To 4000

    For subtr = 0 To 4000 - n

      cube = sumsquares(subtr + n) - sumsquares(subtr)

      If cube = cubes(n) Then

         Text1.Text = Text1.Text & n & Str(subtr + 1) & Str(squares(subtr + 1))

         Text1.Text = Text1.Text & Str(subtr + n) & Str(squares(subtr + n)) & Str(cubes(n)) & crlf

      End If

    Next

  Next

    

  Text1.Text = Text1.Text & crlf & " done"

  DoEvents

End Sub