[seqfan] Re: Pascal triangle & digital root's

allouche at math.jussieu.fr allouche at math.jussieu.fr
Sat Jul 6 14:28:14 CEST 2013


The following remark gives a small hint but does not answer
comletely your question: let p be a prime number, suppose
that the base p expansion of n is a_k a_{k-1} ... a_0,
then {2n \choose n} \equiv {2a_k \choose a_k} {2a_{k-1} \choose a_{k-1}}...
{2a_0 \choose a_0} modulo p.
This is in particular true for p=3.

R.J. McIntosh, A generalization of a congruential property of Lucas,
Amer. Math. Monthly 99 (1992) 231--238,
see also
J.-P. Allouche, Transcendence of formal power series with rational
coefficients, Theoret. Comput. Sci. 218 (1999)143--160.

The above property implies in particular that {2n \choose n} modulo
p is a p-automatic sequence.

[See also the paper by Deutsch and Sagan
http://dx.doi.org/10.1016/j.jnt.2005.06.005) where they prove
that {2n \choose n} is equal to 0 if the base 3 expansion of n
contains a 2, and to (-1)^{\delta_3(n}} if n has only 0's and 1's
where \delta(n) is the number of 1's in the base 3 expansion of n.]


Oh wait a minute, I habe just found an "explicit computation" of the
central binomial coefficients modulo 9 : see a theorem in slides 66--68
at
http://www.mat.univie.ac.at/~kratt/vortrag/3psl2z.pdf

best wishes
jean-paul

Eric Angelini <Eric.Angelini at kntv.be> a écrit :

>
> Hello
>
> A000984 is the "Central binomial coefficients":
> 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432,   
> 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220,   
> 9075135300, 35345263800, 137846528820,...
>
> I was wondering: if we do the digital root of the above terms, do we  
>  see a pattern at some stage?
> Computing the height of Pascal's triangle in digital root's term, I get:
>
> S=1,2,6,2,7,9,6,3,9,2,4,3,7,2,9,9,9,9,6,3,9,3,6,9,9,9,...
>
> ... which is not in the OEIS, and which doesn't show (yet?) any regularity...
>
> Best,
> É.
>
>
>
>
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>
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>





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