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?
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
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Posted by Charlie
on 2017-10-07 22:22:53 |
computer exploration
