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

[Maximilian Hasler]

FWIW, I noticed that the 1st member of the "dominoes" are, e.g.,

A158002 o A000217 = 196*n^2 + 196*n + 1 (n=0,1,2...) for
(1,393), (393,1177), (1177,2353), (2353,3921), (3921,5881), (5881 8233)...,

[Remark: A b-file of 10 000 terms for 1+392k is available... but useful ?]

A035104 o A131877 = 49*n^2 - 42*n - 3 (n=1,2,3,...) for
(4,109), (109,312), (312,613), (613,1012), (1012,1509), (1509,2104)...

etc.: At least all those given earlier are 2nd order polynomials and
can therefore be deduced from the first 3 terms:

polinterpolate( [13,132,349])
%43 = 49*x^2 - 28*x - 8
vector(20,x,subst(%,x,n))
[13, 132, 349, 664, 1077, 1588, 2197, 2904, 3709, 4612, ...]

polinterpolate([24,157,388])
%44 = 49*x^2 - 14*x - 11
vector(20,n,subst(%,x,n))
[24, 157, 388, 717, 1144, 1669, 2292, 3013, 3832, 4749, ...]

Maximilian

On Sat, Jan 4, 2014 at 6:10 AM, Susanne Wienand
<susanne.wienand at gmail.com> wrote:
> 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.
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>
>>>
>>
>
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