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Does it continue? 6: Partial sums (Posted on 2017-09-25) Difficulty: 3 of 5
Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Part A. Write down the positive integers, cross out every second, and form the partial sums of the remaining.

1 2 3 4 5 6 7 8 9 10 11
1   4   9   16  25   36

Does the pattern of squares continue?

Part B. As before, but cross out every third, form partial sums, then cross out every second and for a second partial sums.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 3 7 12 19 27 37 48 61 75 91
1 8 27 64 125 216

Does the pattern of cubes continue?

No Solution Yet Submitted by Jer    
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Some Thoughts re: Notes Comment 7 of 7 |
(In reply to Notes by Jer)

I would expect the algorithm works for all powers, say for tenth powers start with erasing every 10th number, take the partial sums, then erase every 9th number, etc.


To test this I wrote a UBASIC program to try it out.  The output writes the successive sequences with asterisks preceding the values to delete for the next row.
   10   input "B^E B=";B
   20   input "B^E E=";E
   30   Size=E*(B-1)+2
   40   dim V(Size)
   50   for X=0 to (Size-1):V(X)=X:next X
   60   Size=Size-1
   70   for P=E to 2 step -1
   80   J=1
   90   for I=1 to Size
  100   if (I@P)=0 then
  110   :print " *";V(I);
  120   :else print V(I);:V(J)=V(J-1)+V(I):J=J+1
  130   next I
  140   Size=Size-(B-1):print
  150   next P
  160   for X=1 to B:print V(X);:next X

The results for the first five 10th powers:
 1  3  6  10  15  21  28  36  * 45  56  68  81  95  110  126  143  161  * 180  201  223  246  270  295  321  348  376  * 405  436  468  501  535  570  606  643  681  * 720  761 
 1  4  10  20  35  56  84  * 120  176  244  325  420  530  656  799  * 960  1161  1384  1630  1900  2195  2516  2864  * 3240  3676  4144  4645  5180  5750  6356  6999  * 7680  8441 
 1  5  15  35  70  126  * 210  386  630  955  1375  1905  2561  * 3360  4521  5905  7535  9435  11630  14146  * 17010  20686  24830  29475  34655  40405  46761  * 53760  62201 
 1  6  21  56  126  * 252  638  1268  2223  3598  5503  * 8064  12585  18490  26025  35460  47090  * 61236  81922  106752  136227  170882  211287  * 258048  320249 
 1  7  28  84  * 210  848  2116  4339  7937  * 13440  26025  44515  70540  106000  * 153090  235012  341764  477991  648873  * 860160  1180409 
 1  8  36  * 120  968  3084  7423  * 15360  41385  85900  156440  * 262440  497452  839216  1317207  * 1966080  3146489 
 1  9  * 45  1013  4097  * 11520  52905  138805  * 295245  792697  1631913  * 2949120  6095609 
 1  * 10  1023  * 5120  58025  * 196830  989527  * 2621440  8717049 
 1  1024  59049  1048576  9765625 

Edited on October 5, 2017, 2:16 pm
  Posted by Brian Smith on 2017-10-05 14:14:49

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