Nothing new under the sun

[Richard Guy]

Thankyou!  I'm copying it to the
original addressees.    R.

On Thu, 30 Mar 2006, Robert G. Wilson v wrote:

> Dear Sir,
>
> 	But they are not identical.
>
> 	A025480 uses the natural numbers for a(2n)=n
> whereas A108202 a(2n)=A007376(n): The almost-natural
> numbers: write n in base 10 and juxtapose digits.
>
> 	This then produces two different sequences.
> Let me place them one over the other.
>
> 0, 0, 1, 0, 2, 1, 3, 0, 4, 2, 5, 1, 6, 3, 7, 0, 8, 4, 
> 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 0, 16, 
> 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 
> 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 
> 31, 0, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 
> 38, 19, 39, 2, 40, 20, 41, 10
> 0, 0, 1, 0, 2, 1, 3, 0, 4, 2, 5, 1, 6, 3, 7, 0, 8, 4, 
> 9, 2,  1, 5,  0, 1,  1, 6,  1, 3,  1, 7,  2, 0,  1, 
> 8,  3, 4,  1, 9,  4, 2,  1,  1,  5, 5,  1,  0,  6, 1, 
> 1,  1,  7, 6,  1,  1,  8, 3,  1,  1,  9, 7,  2,  2, 
> 0, 0
>
> 	I hope this clarifies things.
>
> Sincerely yours,
>
> Robert G. 'Bob' Wilson, V
> ATP / CF & GI
>
> Richard Guy wrote:
>> Sequences A025480 and A108202 are identical
>> and much older than the references given
>> there.  They are the nim-values for heaps of
>> n  beans in the game where you're allowed to
>> take up to half of the beans in a heap.
>> 
>> I can't quote chapter and verse, but this
>> has been well known to those who well know
>> it for half a century.     R.
>> 
>> 
>