[seqfan] Re: A new sequence in Dutch magazine "Pythagoras"
jonscho at hiwaay.net
Fri Dec 2 01:30:12 CET 2016
I don't know Dutch, but copying the text from the first page (i.e., the one
numbered 20, but it's page 22 of the PDF) into the Bing Translator and
attempting to correct the grammar in its output (and taking some guesses in
places where that output it made very little sense to me), I got the
NUMBER SEQUENCES episode 11
In the September issue we wrote a competition as part of our series on
integer sequences. In this latest issue of volume 55 we give the result.
by Matthijs Coster
Keep in mind that a best original integer sequence still has not appeared in
the Online Encyclopedia of Integer Sequences (OEIS). That was the Mission of
the price question we wrote in September. Also there was a New Year's
challenge: create a sequence containing the number 2016. We wrote about this
bonus question in the February issue.
The most beautiful entry – without the restriction that 2016 must be in it –
according to the jury was an integer sequence from Pim Sailor (Gymnasium
Sorghvliet, The Hague). It is the sequence
1, 3, 5, 8, 11, 14, 17, 21, 24, 28, 32, 36, 40, 44, 49, 53, 57, 62, 66,
71, 75, 80, 84, 90, 94, 99, ...
If you do not see the system behind it as soon as you see this sequence,
don't worry, because what Pim has figured out is certainly resourceful. Pim
based it on a variant on the game Battleship. The sea is divided into n
areas. You have n^2 ships available to you that can apportion among these n
areas. Your opponent may look at the way you have apportioned your ships and
then decide to completely bomb any one area. All ships in that area are then
destroyed. If you are smart, you'll spread the ships as evenly as possible
among the n areas. However, in your first move, you will lose n ships. You
will then reapportion the remaining ships among the n areas. What Pim
wondered was: how many bombings are required in order to destroy all ships?
Let's look at an example with n = 5 (see illustration). The sea is divided
into 5 areas in which your 25 ships must be apportioned. The sequence of
numbers of remaining ships is as follows: 25, 20, 16, 12, 9, 7, 5, 4, 3, 2,
1, 0. After 11 bombings, all ships are destroyed. So, a_5 = 11.
Can we predict the numbers in Pim's sequence? That works pretty reasonably.
Let's begin again with n areas. It is quite difficult to determine the exact
value of a_n but we can give an approximation. Let us begin, not with n^2
ships, but with n ships. It is clear that only after n bombings are all
ships destroyed. If we start with 2n ships, then after the first (1/2)n
bombings, n ships are destroyed.
I was going to go on and try to (very roughly!) translate the next page, but
I realized at the bottom of the above (very rough!) translation of the first
page that I didn't understand the last paragraph above. If n=5 and we start
with 2n = 10 ships, what does it mean to say that "after the first (1/2)n
bombings, n ships are destroyed"? "The first (1/2)n bombings" are the first
2.5 bombings, and it's hard to have a nonintegral number of bombings! :-)
(I think what it means is that Bing Translator and I together did not come
up with a reliable translation....) :-(
----- Original Message -----
From: "Neil Sloane" <njasloane at gmail.com>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Thursday, December 01, 2016 2:00 PM
Subject: [seqfan] A new sequence in Dutch magazine "Pythagoras"
> Dear Sequence Fans, In the June 2016 issue of Pythagoras
> on pages 20 and 21 there is a new sequence from Pim Spelier.
> See http://www.pyth.eu/jaargangen/Pyth55-6.pdf
> It looks like Pythagoras had a competition to
> find the best sequence not in the OEIS,
> and this was the winner. I think we should
> add it to the OEIS, but my Dutch is not good enough
> to give a clear definition.
> Maybe someone who speaks Dutch would be willing to submit the sequence (if
> so, please tell me the A-number).
> Thanks to Omar Pol who found the magazine (there is also an article about
> the toothpick sequence A139250).
> Seqfan Mailing list - http://list.seqfan.eu/
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