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Does it continue? 10: Alternate differences (Posted on 2017-10-07) 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."

Consider the sequence

a1=1, an+1=[√(2an(an+1))] for n≥1 where [x] is the floor function.

Here are the first 17 terms and the alternate differences a2k+1 - a2k

1 2 3 4 6 9 13 19 27 38 54 77 109 154 218 309 437
   1   2   4     8    16    32      64      128
Are they all powers of 2?

No Solution Yet Submitted by Jer    
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Some Thoughts computer exploration Comment 2 of 2 |
The first 49 generations, showing the position of the larger number in the subtraction, the lower and larger numbers themselves, the difference, the power of 2 that is represented (i.e., the base-2 log of the difference--which turns out to be an integer), and the fractional portion that was lopped off in taking the floor of the larger number in the difference. This last was included to give an idea of the effects of this truncation. There seems to be no secular or asymptotic change, just seemingly random values. They're constrained to be between zero and 1, but what if in some instance the power of 2 required a difference outside this range? 

3     2 3    1 0     0.464101615137754
5     4 6    2 1     0.324555320336759
7     9 13    4 2     0.416407864998739
9     19 27    8 3     0.568097504180443
11     38 54    16 4     0.442630355264797
13     77 109    32 5     0.599270070562056
15     154 218    64 6     0.494851197917257
17     309 437    128 7     0.698526385456319
19     618 874    256 8     0.690802512522168
21     1236 1748    512 9     0.674926908943917
23     2472 3496    1024 10     0.642961470330192
25     4944 6992    2048 11     0.578923401580141
27     9888 13984    4096 12     0.450793649351908
29     19777 27969    8192 13     0.608720895615988
31     39554 55938    16384 14     0.510348417396017
33     79108 111876    32768 15     0.313596757376217
35     158217 223753    65536 16     0.334303647861816
37     316435 447507    131072 17     0.375715752888937
39     632871 895015    262144 18     0.458539124927484
41     1265743 1790031    524288 19     0.624185449909419
43     2531486 3580062    1048576 20     0.541264328174293
45     5062972 7160124    2097152 21     0.375421980395913
47     10125945 14320249    4194304 22     0.457950793206692
49     20251891 28640499    8388608 23     0.623008396476507
51     40503782 57280998    16777216 24     0.538910023868084
53     81007564 114561996    33554432 25     0.370713263750076
55     162015129 229123993    67108864 26     0.448533326387405
57     324030259 458247987    134217728 27     0.604173421859741
59     648060518 916495974    268435456 28     0.501240015029907
61     1296121036 1832991948    536870912 29     0.295373201370239
63     2592242073 3665983897    1073741824 30     0.297853469848633
65     5184484147 7331967795    2147483648 31     0.302813529968262
67     10368968295 14663935591    4294967296 32     0.312734603881836
69     20737936591 29327871183    8589934592 33     0.332572937011719
71     41475873183 58655742367    17179869184 34     0.37225341796875
73     82951746367 117311484735    34359738368 35     0.451614379882813
75     165903492735 234622969471    68719476736 36     0.6103515625
77     331806985470 469245938942    137438953472 37     0.5135498046875
79     663613970940 938491877884    274877906944 38     0.320068359375
81     1327227941881 1876983755769    549755813888 39     0.34716796875
83     2654455883763 3753967511539    1099511627776 40     0.4013671875
85     5308911767527 7507935023079    2199023255552 41     0.509765625
87     10617823535054 15015870046158    4398046511104 42     0.3125
89     21235647070109 30031740092317    8796093022208 43     0.33203125
91     42471294140219 60063480184635    17592186044416 44     0.375
93     84942588280439 120126960369271    35184372088832 45     0.453125
95     169885176560879 240253920738543    70368744177664 46     0.625
97     339770353121758 480507841477086    140737488355328 47     0.5
99     679540706243516 961015682954172    281474976710656 48     0.375

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 a = 1
 For gen = 2 To 200
    b0 = Sqr(2 * (a * (a + 1)))
    b = Int(b0)
    fract = b0 - b
    If gen Mod 2 = 1 Then
      diff = b - a
      l2 = Log(diff) / Log(2)
     ' If l2 <> Int(l2) Then
       Text1.Text = Text1.Text & gen & "     " & a & Str(b) & "    " & diff & Str(l2) & "     " & fract & crlf
     ' End If
    End If
    a = b
    DoEvents
 Next gen
 
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub



  Posted by Charlie on 2017-10-07 22:22:53
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