[seqfan] Re: Higher dimensional analogues to plane division by ellipses?

[Jess Tauber]

Note added: The Fib side boils down to 2xLuc, and the Luc side boils down
to 6xFib, except for the first couple of terms in the latter case.


On Sat, Jun 1, 2013 at 5:12 AM, Jess Tauber <yahganlang at gmail.com> wrote:

> In the hours since I first posted, I worked out that A051890 consists of
> terms which are all 4TRI+2 (TRI=triangular number) while A014206 uses
> 2TRI+2.
>
> I guess something clicked in the back of my head, because later on I
> back-engineered a generalized Pascal Triangle with A051890 as the analogue
> for the doubled triangular number diagonal.
>
> Lo and behold A014206 occupies the OTHER doubled triangular number
> diagonal.
>
> Furthermore, the resulting Pascal system is a variation on the simple
> (2,1)-sided version, which has Lucas numbers on one side and Fibonacci on
> the other as the summations over the shallow diagonals.
>
> Here, however, while the Triangle itself has doubled values within, the
> Fib and Luc numbers are SINGLE, with Fib increments. I'm sure there are
> many ways to cut the numbers here so they are related to doubled values in
> other combinations.
>
> The (2,1)-sided version of the Pascal Triangle appears to be special in at
> least this- diagonal values are identical to numerical coefficients of
> power terms that sum to powers of Metallic Means (and the dimensional
> labels of the diagonals are the same as the powers the terms are raised
> to)- I worked this out about a year and a half ago and still haven't yet
> found out if this is an old result, though some work by Koshy appears to
> suggest it is known).
>
> This also relates to atomic nuclei, in that the ratios of neutrons to
> protons (also taking into account stability) seems to follow a smooth curve
> of what might be the Metallic Means if fractional values were allowed as
> well into the equations describing them. Physicists noticed a relation to
> Phi as far back as 1916, well before neutrons were even discovered, based
> on atomic masses.
>
> Here is the Triangle in question:
>
> 2
> 2,2
> 4,4,2
> 4,8,6,2
> 4,12,14,8,2
> 4,16,26,22,10,2
> 4,20,42,48,32,12,2
> 4,24,62,90,80,44,14,2
> 4,28,86,152,170,124,58,16,2
> 4,32,114,238,322,294,182,74,18,2
>
> The Luc side of the Triangle has (in descending order):
>
> 322+8=330, 199+5=204, 123+3=126, 76+2=78, 47+1=48, 29+1=30, 18+0=18,
> 11+1=12, 7-1=6, 4+0=4, 3-1=2, 1+1=2
>
> The Fib side has (again descending):
>
> 233+13=246, 144+8=152, 89+5=94, 55+3=58, 34+2=36, 21+1=22, 13+1=14, 8+0=8,
> 5+1=6, 3-1=2, 2+0=2
>
> All in all this is a very odd result. Unlike the situation for the atomic
> electronic system (where Luc and Fib relations become obvious when the
> numbers are taken as atomic numbers, in terms of their quantum number
> preferences), nothing of the sort had been visible til now for the nuclear
> system. And it was WELL hidden.
>
> Jess Tauber
>
>
> On Fri, May 31, 2013 at 3:47 PM, Jess Tauber <goldenratio at earthlink.net>wrote:
>
>> OEIS A051890 relates to the maximal number of regions into which
>> ellipsoids may divide the plane. A014206 for division of the plane by
>> circles.
>>
>> I've seen BOTH these sequences with regard to the combinatorics of
>> nucleons in real nuclei.
>>
>> All sorts of questions come to mind, but I'll limit myself to these:
>>
>> Firstly, are there analogous sequences for higher dimensional analogues,
>> say, volume elements in a larger volume cut by ellipsoids or spheres, and
>> are the formulae regularly relatable to those of the plane?
>>
>> Secondly, what if we consider other conic sections (and higher
>> analogues), either uniformly or in combination?
>>
>> I tried looking up volume division at the Wolfram site, no luck.
>>
>> Thanks.
>>
>> Jess Tauber
>> goldenratio at earthlink.net
>>
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>
>