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Does it continue? 7: Partial sums again (Posted on 2017-09-28) Difficulty: 3 of 5
Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Write down the positive integers. Take the first number. Delete the second after that, the third after that, and so on. Form the partial sums and repeat.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
2 6 11 18 26 35 46 58 71 85 101 118 136 155 175
6 24 50 96 154 225 326 444 580 735
24 120 274 600 1044 1624
120 720 1764
720

Does the pattern of factorials continue?

No Solution Yet Submitted by Jer    
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Some Thoughts computer exploration Comment 1 of 1
The truncated table below confirms the suspicion the factorials continue:

 2     6     11     18     26     35     46     58     71    
 6     24     50     96     154     225     326     444     580    
 24     120     274     600     1044     1624     2556     3708     5104    
 120     720     1764     4320     8028     13132     22212     33984     48860    
 720     5040     13068     35280     69264     118124     212976     341136     509004    
 5040     40320     109584     322560     663696     1172700     2239344     3733920      
 40320     362880     1026576     3265920     6999840     12753576     25659360       
 362880     3628800     10628640     36288000     80627040     150917976     318540960      
 3628800     39916800     120543840     439084800     1007441280     1931559552        
 39916800     479001600     1486442880     5748019200     13575738240     26596717056  
 479001600     6227020800     19802759040     80951270400     196287356160     392156797824   
 6227020800     87178291200     283465647360     1220496076800     3031488633600     
 87178291200     1307674368000     4339163001600     19615115520000     49811492505600
 1307674368000     20922789888000     70734282393600     334764638208000     867718162483200 
 20922789888000     355687428096000     1.2234055905792E+15     6.046686277632E+15     


(rows truncated to keep the table from stretching too far to the right)

DefDbl A-Z
Dim crlf$, sums(1, 10000)


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 lim = 10000
 For i = 1 To lim
   sums(0, i) = i
 Next
 For generation = 1 To 15
 avoid0 = 1: avoid = 1: take = 1
 For i = 1 To lim
   If take = avoid Then
     avoid0 = avoid0 + 1
     avoid = (avoid0 * avoid0 + avoid0) / 2
     take = take + 1
   End If
   sums(1, i) = sums(1, i - 1) + sums(0, take)
   take = take + 1
   If take > lim Then lim = i - 1: Exit For
   If i < 10 Then
      Text1.Text = Text1.Text & Str(sums(1, i)) & "    "
   End If
   DoEvents
 Next
 Text1.Text = Text1.Text & crlf
 For i = 1 To lim
    sums(0, i) = sums(1, i)
 Next
 Next generation
 
 Text1.Text = Text1.Text & crlf & crlf
 


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


  Posted by Charlie on 2017-09-28 15:15:27
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