[seqfan] Re: Minimal k > n such that (4k+3n)(4n+3k) is square

[Susanne Wienand]

Dear seqfans,

'The same numbers can be reached twice' is expressed ambigously. I mean
'There are cases where the same numbers are reached twice'.
Another example is:

(14101,*45496*)

(1221,5176), (5176,11581), (11581,20436), (20436,31741), (31741, *45496*), (
*45496*,61701)...

Regards
Susanne






2013/12/27 Susanne Wienand <susanne.wienand at gmail.com>

> Dear seqfans,
>
> other ratios of coprime (n,k) which fulfil (3k+4n)(4k+3n) = square number
> can be appended to (1,393), (4,109), (13,132), (24,157) etc like dominoes:
>
> (1,393), (393,1177), (1177,2353), (2353,3921), (3921,5881), (5881 8233)...
>
> (4,109), (109,312), (312,613), (613,1012), (1012,1509), (1509,2104)...
>
> (13,132), (132,349), (349,664), (664,1077), (1077,1588), (1588,2197)...
>
> (24,157), (157,388), (388,717), (717,1144), (1144,1669), (1669,2292)...
>
> The same numbers can be reached twice:
>
> (321,7153), (7153,20257)
>
> (769,3289), (3289,7377), (7377,13033), (13033,20257)...
>
> Regards
> Susanne
>
>
>
>
>
>
>
> 2013/12/26 Kevin Ryde <user42 at zip.com.au>
>
>> charles.greathouse at case.edu (Charles Greathouse) writes:
>> >
>> > If there is no n < k < 109n/4 with (4k+3n)(4n+3k) square, then a(n) =
>> 393n.
>>
>> I tried continuing past 393 where there's further multipliers
>>
>>     f=393, 76441, 14829361, 2876819793
>>
>> which is solutions to 12*f^2+25*f+12=square, excluding multiples of
>> earlier solutions, and which therefore k=f*n gives (4k+3n)(4n+3k)=square
>> for any n.
>>
>> It seems when a particular n allows f=109/4=27.25 that there's a single
>> extra solution between the integer ones.  Eg. n=4
>>
>>     f = 109/4 = 27.2500
>>         393
>>         21949/4 = 5487.2500
>>         76441
>>         4258797/4 = 1064699.2500
>>
>> But when there's more than one solution they're in pairs.  Eg. n=13
>>
>>     f = 10.1538 = 132/13         \ new pair
>>         70.4615 = 916/13         /
>>         393
>>         2170.4615 = 28216/13        \ new pair
>>         13870.1538 = 180312/13      /
>>         76441
>>         421259.3846 = 5476372/13      \ new pair
>>         2690939.3846 = 34982212/13    /
>>         14829361
>>
>> It's possible to have a mixture of 27.25 and pairs.  Eg. n=24
>>
>>     f = 157/24 = 6.5417                <----+
>>         654/24 = 27.2500    <- single       |- pair
>>         2509/24 = 104.5417             <----+
>>         393
>>         35269/24 = 1469.5417
>>         131694/24 = 5487.2500
>>         491557/24 = 20481.5417
>>         76441
>>
>> Dunno if this is always so or if it has any significance.
>>
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>>
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>>
>
>