One famous series
One famous series in mathematical puzzles' paraphilology is the following:
1
11
21
1211
111221
Eric Weisstein describes this as the Look and Say sequence
http://mathworld.wolfram.com/news/2004-10-13/google/, item 3)
In his view, the next term is
312211
The problem becomes more challenging when one notices that for the given terms
of the series, the sum of the numbers in each term gives the terms of the Fibonacci
series from the second one on, namely 1,2,3,5,8,13,21 and so on and so forth ad vitam aeternam.
I divide the terms of the Look and Say series in pairs and proceed to derive the next term
in the series by Looking and Saying each pair separately.
So the series would go like this, in my book:
1
11
21
12 | 11
11 | 12 | 21
21 | 11 | 12 | 12 | 11
12 | 11 | 21 | 11 | 12 | 11 | 12 | 21
11 | 12 | 21 | 12 | 11 | 21 | 11 | 12 | 21 | 11 | 12 | 12 | 11
and so on and so forth, ad vitam aeternam.
The sum of the numbers in each term of the series is the (n+1)th Fibonacci
number.
From the third term of this series on,
1) Taking f(n) pairs from the beginning of the term where f(n)
is the nth Fibonacci number (f(n)=1 for the third term, f(n)=1
for the fourth term, f(n) =2 for the fifth term and so on) we reach
a division point where the sum of the numbers on the left hand side
equals f(n+2) and the sum of the numbers on the right hand
side equals f(n+1).
2) The number of pairs in the series recapitulates the sum of the
numbers in the series from the first term on.
3) The number of twos in the series gives the Fibonacci series.
4) The number of ones in the series gives the Lucas series
(1,3,4,7,11,18,29...)
Now isn't that beautiful and original?
Dimitri Boscainos
Translator and Amator Mathesi
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