Re: Analyzing sequences of fractions in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg93643] Re: Analyzing sequences of fractions in Mathematica
• From: "Steve Luttrell" <steve at _removemefirst_luttrell.org.uk>
• Date: Thu, 20 Nov 2008 04:55:28 -0500 (EST)
• References: <200811130904.EAA14027@smc.vnet.net> <gfimdm\$fsc\$1@smc.vnet.net>

```Is your nice solution obtained using the "Integer Sequence Analysis" methods
that have just been added to Mathematica 7
(http://www.wolfram.com/products/mathematica/newin7/content/IntegerSequenceAnalysis/)?

--
Stephen Luttrell
West Malvern, UK

"Eric W. Weisstein" <eww at wolfram.com> wrote in message
news:gfimdm\$fsc\$1 at smc.vnet.net...
> sergio_r at mail.com wrote:
>
>> Hello Mathematica users,
>>
>> (* I am trying to infer, if possible, the general term
>>    of the following sequence: *)
>>
>> mysequence = {1/8, 3/16, 11/32, 17/64, 25/128, 59/256, 147/512,
>>  265/1024, 465/2048,
>>  995/4096, 2171/8192, 4161/16384, 7881/32768, 16203/65536,
>> 33571/131072,
>>  65977/262144, 129025/524288, 260979/1048576, 529547/2097152,
>>  1051505/4194304, 2083705/8388608, 4186715/16777216, 8423091/33554432,
>>  16796521/67108864, 33466161/134217728, 67059203/268435456,
>>  134443931/536870912}
>
> It can probably be simplified somewhat but here is one form of it:
>
> seq[n_] := With[{
>       a = 1/2 (-1 + Sqrt[17]), b = 1/2 (1 + Sqrt[17]),
>       c = Sqrt[1/17 (-4 + Sqrt[17])], d = Sqrt[1/17 (4 + Sqrt[17])]
>     },
>     2^(-n - 2) Piecewise[{
>       {2^n - a^(n/2)/Sqrt[17] + (I^n b^(n/2))/Sqrt[17], Mod[n, 2] == 0},
>       {2^n + c a^(n/2) + I^(n + 1) d b^(n/2), Mod[n, 2] == 1}
>    }]]
>
> Table[seq[n], {n, 27}] // FullSimplify
> {1/8, 3/16, 11/32, 17/64, 25/128, 59/256, 147/512, 265/1024, \
> 465/2048, 995/4096, 2171/8192, 4161/16384, 7881/32768, 16203/65536, \
> 33571/131072, 65977/262144, 129025/524288, 260979/1048576, \
> 529547/2097152, 1051505/4194304, 2083705/8388608, 4186715/16777216, \
> 8423091/33554432, 16796521/67108864, 33466161/134217728, \
> 67059203/268435456, 134443931/536870912}
>
> % == mysequence
> True
>
> Cheers,
> -Eric
>

```

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