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Double a triangle (Posted on 2015-02-19) Difficulty: 3 of 5
In the sequence of triangular numbers, some numbers are twice another.

For example t(20)=210 which is twice t(14)=105.

Characterize all such numbers.

Easy bonus: Explain why (except for the trivial case) there are no square numbers that are twice another.

No Solution Yet Submitted by Jer    
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Some Thoughts The first few pairs | Comment 1 of 3
If n is the ordinality of the smaller triangular number and x the ordinality of the larger, then

x*(x+1) = 2*n*(n+1)

Treating this as a quadratic in x for the given n, we need a positive value of x

(- 1 + sqrt(1 + 8*n^2 + 8*n)) / 2

When this is an integer we have a valid case.

The first two numbers in each line below are the ordinalities and the next two are the actual triangular numbers:

2 3     3 6
14 20     105 210
84 119     3570 7140
492 696     121278 242556
2870 4059     4119885 8239770
16730 23660     139954815 279909630
97512 137903     4754343828 9508687656
568344 803760     161507735340 323015470680
3312554 4684659     5486508657735 10973017315470


DefDbl A-Z
Dim fct(20, 1), crlf$
Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 
  For n = 1 To 10000000
    disc = 1 + 8 * n * n + 8 * n
    sr = Int(Sqr(disc) + 0.5)
    If sr * sr = disc Then
     If sr Mod 2 = 1 Then
       x = (sr - 1) / 2
       Text1.Text = Text1.Text & n & Str(x) & "     " & tr(n) & Str(tr(x)) & crlf
       DoEvents
     End If
    End If
  Next
 

  Text1.Text = Text1.Text & "done"
  DoEvents

End Sub
Function tr(x)
  tr = x * (x + 1) / 2
End Function

The lower triangular number appears as sequence A075528 in Sloane's OEIS, the higher being A029549, with ordinalities A053141 and A001652 respectively.

  Posted by Charlie on 2015-02-19 14:07:43
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