known (coincidental ?) artifact ?

[Alexander Povolotsky]

Thanks for the reply,  Franklin,

Could you also explain why the pattern (at least up to n=60) keeps steadily
alternating between 4 terms and 6 terms in the each two consecutive groups,
with the difference remaining the same within each such 4-term or
6-term group (while during the switch from the 4-group to the 6-group and
then back to the next 4-group, etc. the difference is getting bumped by 1)
?
How you explain the fact that the sum of number of terms for each two
consecutive groups is  4+ 6= 10 ?

Note that if the pattern indeed could be described as discuseed in the
predictable manner,
then one could resolve the iteration and derive the following formula (non
trivial identity ?)

n = (floor(n*n/10)+10 - someFunction(n) ) /2

Thanks,
Alexander R. Povolotsky

On 1/11/08, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> It's pretty much obvious.
>
> First of all, simple arithmetic lets us simplify the expression to
>
> a(n) = floor(n^2/10) - floor((n-1)^2/10) + 2
>
> The "+ 2" obviously doesn't affect the pattern of differences,
> so what we are really looking at is the pattern of differences in
> floor(n^2)/10.   Since the final digit of n^2 depends only
> on n modulo 10, and the difference in the squares is 2n-1, this
> will have a pattern of changes depending only on the final
> digit.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Alexander Povolotsky apovolot at gmail.com
> .
> Consider calculating (in decimal system):
>
> floor(n*n/10) + 10
>
> and then looking between the  difference of calculated above and
> "doubled" value of n :
>
> (floor(n*n/10) + 10)  - 2*n
>
> and then look at the differences between in above expression for two
> consecutive integers n and (n-1)
>
> a(n) = ((floor(an*n/10) + 10) - 2*n) - ((floor((n-1)*(n-1)/10) + 10 -
> 2*(n-1))
> ...
>
> So you could notice that starting from a(4) and on - the difference is
> kept the same for first 4 terms and then the difference gets
> incremented by 1 and as such is kept constant for next 6 terms, then it
> increments by 1 again and is kept the same for 4 terms, etc, etc ....
> ...
> Is this well known and/or obvious ?
> ____________________________________________________________________
>
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