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

[Eric Angelini]

Thanks for the .pdf, Jean-Paul !
Best,
É.

Propulsé d'un aPhone



Le 6 juil. 2013 à 14:28, "allouche at math.jussieu.fr" <allouche at math.jussieu.fr> a écrit :

> 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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