Bottema Article on Malfatti Problem
Antreas P. Hatzipolakis
xpolakis at otenet.gr
Mon Jan 10 17:57:14 CET 2000
David W. Wilson writes:
>"Antreas P. Hatzipolakis" wrote:
>>
>> Oene Bottema (1901-1992), the great Dutch geometer, published several
>> articles in the Dutch periodical _Euclides_. Among them is one on Malfatti
>> Problem:
>>
>> O. Bottema: Verscheidenheden XXVI. Het vraagstuk van Malfatti.
>> Euclides 25 (1949-50), pp. 144-149. [in Dutch]
>>
>> Jan P. Hogendijk kindly sent me a photocopy of the article, which I typed
>> and put in my web pages. Its URL is:
>>
>> http://users.hol.gr/~xpolakis/bottema/malfatti.html
>>
>> Bottema generalizes Malfatti Problem, and gives interesting sequences
>> of circles. In these sequences appear two recursive integer sequences:
>>
>> ------
>>
>> a(n): 0,16,160,1600,15840,....
>>
>> a(1) = 0, a(2) = 16
>> a(2p+1) = 10a(2p) - a(2p-1)
>> a(2p) = 10a(2p-1) - a(2p-2) + 16
>>
>> ------
>>
>> b(n): 2, 20, 198, 1960, .....
>>
>> b(1) = 2, b(2) = 20
>> b(k) = 10b(k-1) - b(k-2)
>>
>> Antreas
>
>a is new, b is A001078.
In the sequence's entry in EIS
( http://www.research.att.com/~njas/sequences/SA.html )
we read:
References V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques.
Gauthier-Villars, Paris, 1952, p. 281.
Victor Thebault was a distinguished French geometer, so most likely
he refers to Bottema's paper, where the sequence is found.
Antreas
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