[seqfan] Skepticism on A271650 and thereafter
Brad Klee
bradklee at gmail.com
Mon May 27 00:49:53 CEST 2019
Hi Seqfans,
Cristoph Koutschan has entered some nice sequences on
"higher dimensional f.c.c. lattices" that extend upon the
d-dimensional Polya integrals ( cf. A287318, [1] ).
Calculating integrand reductions over a twelve-dimensional
vector space, I verified in about dt=60(s) the following
differential equation:
> https://oeis.org/A271650
> > http://www.koutschan.de/data/fcc1/fcc5_mop.txt
We now have two rigorous and logically independent
algorithms outputting the same result; however, I am
skeptical as to the minimality of either calculation.
The sextic polynomial:
-675000+3465000*x-1053375*x^2+933650*x^3
+449735*x^4+144776*x^5+15678*x^6 = 0
indicates an pole x=0.205..., well within the integral's
convergence region [0,1). Can this be explained away
in terms of complex geometry? Or does anyone have
a D.E. with lower order 5?
For higher dimensions, according to
> https://oeis.org/A271674
> > http://www.koutschan.de/data/fcc1/
the predicted order 6 is part of an integer sequence
(something akin to a genus-degree formula) with first
few terms (from d=2):
a(d) : 2,3,4,6,8,11,14,18,22,27,32 . . .
Are these numbers from guessing reliable? If yes,
is the overlap with A290743 more than coincidence?
This looks to be an intriguing and even plausible
possibility, especially considering that the count of
walk generators grows like n^2. This could explain
difference from the Polya case, where the pattern
starting 2,3,4 continues linearly with 5,6,7,8 . . .
--Brad
[1] http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
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