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

[Susanne Wienand]

Hello seqfans,

A085018 <http://oeis.org/A085018> seem to be the starting n-values of the
'dominoes' (1,4,13,24,33,37,52,61,...) and A085019
<http://oeis.org/A085019>seem to be the corresponding values of k
(393,109,132,157,481,184,213,1048,...).

Regards
Susanne


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

> 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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>>>
>>
>>
>