[seqfan] Re: A216607, A232091.
David Applegate
david at research.att.com
Fri Jan 10 22:02:06 CET 2014
Ed Jeffrey asked:
> The formula for A216607 is
> a(n) = floor((1/4)*ceil(sqrt(4*n))^2) - n (n>0),
> and the formula for A232091 is
> b(n) = ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n)) (n>0).
> Could someone please (try to) find the least n for which
> (1) a(n) = b(n) - n
> does not hold and let me know which n it is?
The equation (1) is true.
Here is an ugly proof. I suspect that there is a much simpler proof,
probably using standard results about integer square roots I don't
know, but I just bashed along.
We want to show (for all integers n > 0):
floor((1/4)*ceil(sqrt(4*n))^2) == ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n))
Write sqrt(n) = k - f, where k is an integer and 0 <= f < 1, so:
ceiling(sqrt(n)) = k
n = k^2 - 2kf + f^2
floor((1/4)*ceil(sqrt(4*n))^2) = floor((1/4)*ceil(2k-2f)^2)
ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n)) = ceiling(n/k)*k
= ceiling(k - 2f + f^2/k) * k
First, consider the case when 0 <= f < 1/2. In this case,
floor((1/4)*ceil(2k-2f)^2) = floor(k^2) = k^2
Since k >= 1, f^2/k <= 2f, so
0 <= 2f - f^2/k < 1
and
ceiling(k - 2f + f^2/k) * k = k^2
and we are done.
Now, consider the case when 1/2 <= f < 1. In this case,
floor((1/4)*ceil(2k-2f)^2) = floor(k^2 - k + 1/4) = k^2 - k
Note that
n - k^2 + k = f^2 - (2f-1) k
The left hand side is an integer, and the right hand side is < 1, so
it must be <= 0. But
f^2 - (2f-1) k <= 0
implies
f^2/k - 2f <= -1
or
2f - f^2/k >= 1
and since
2f - f^2/k < 2,
we have
ceiling(k - 2f + f^2/k) * k = (k-1)*k = k^2 - k
and we are done.
David Applegate
> From seqfan-bounces at list.seqfan.eu Thu Jan 9 14:19:22 2014
> From: "Lars Blomberg" <lars.blomberg at visit.se>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Thu, 9 Jan 2014 18:24:01 +0100
> Subject: [seqfan] Re: A216607, A232091.
> The equation (1) is true for all n < 10^9.
> Regards
> /Lars Blomberg
> -----Ursprungligt meddelande-----
> From: L. Edson Jeffery
> Sent: Thursday, January 09, 2014 6:02 AM
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] A216607, A232091.
> The formula for A216607 is
> a(n) = floor((1/4)*ceil(sqrt(4*n))^2) - n (n>0),
> and the formula for A232091 is
> b(n) = ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n)) (n>0).
> Could someone please (try to) find the least n for which
> (1) a(n) = b(n) - n
> does not hold and let me know which n it is?
> If (1) is not true, then I want to submit the sequence {b(n) - n} since it
> is related to A232091.
> Thanks,
> Ed Jeffery
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