[seqfan] self-cooperation and the EIS

[Olivier Gerard]

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One of my mathematical mentors, Pierre Cartier, use to say:

"On ne collabore jamais assez avec soi-m=EAme"

and he has several anecdotes where has been searching for
an insight he might have found in his own works published
years ago. He has also analyzed how several scientists have
failed to read themselves at important moments in their research.

To me one of the interest of the EIS, (thanks Neil!) is that not
only it represents (as well as the ISC) a shared repository of accumulated
wisdom by generations of mathematicians but also of one's personal
investigations. And sometimes two puzzle pieces fit together.

A very recent but minor example (2 minutes ago).
Producing a few diagonals to test ideas which came from responding
to Wouter, I asked the EIS for this sequence (among several others):

1,5,27,194,1865,22875,...


Matches (up to a limit of 10) found for  1 5 27 194  :

%I A023811
%S A023811 1,5,27,194,1865,22875,342391,6053444,123456789,2853116705,
%T A023811 73686780563,2103299351334,65751519677857,2234152501943159,
%U A023811 81985529216486895,3231407272993502984,136146740744970718253
%N A023811 Largest Metadrome (number with digits in strict ascending order)
in base n.
%R A023811
%O A023811 2,2
%K A023811 nonn,easy
%A A023811 Olivier Gerard (quadrature at onco.techlink.fr)

I am always amused to discover a symptom of what I might call
a personal recurrence. The formula I used to compute the test
sequence can be simplified into:

      Sum[ Binomial[n,j] (n-1)^(n-2-j), {j=3D0->n-2}]

Of course, this example is quite useless, since I had beforehand a
much simpler formula from the definition:

      Sum[ j (n)^(n-1-j), {j=3D1->n-1}]

So I have simply rediscovered a basic application of a basic
combinatorial identity.  Not much, really. (I have often
been much more lucky and I bet you too. Can we share on
this mailing list a few examples ?)

Anyway, as some of the other sequences I tested were not in the database, I
will propose them and look out if they do not have, by chance, a
nice digit pattern interpretation, a thing I would not have thought about
without this event. But perhaps Patrick "palindrome" DeGeest is interested ?


Olivier