[seqfan] Fun little identity (to me anyway)
David Wilson
dwilson at gambitcomm.com
Wed Jan 21 15:32:35 CET 2009
While lying in bed last night, I kept myself occupied working on a sequence
a(1) = 1;
a(n+1) = smallest value > a(n) with a(n)a(n+1) of form k^2-2.
I got as far as
1,2,7,14,23,34,...
when I realized that after 1, these numbers were themselves of the form
k^2-2. I was able to formulate
(n^2-2)((n+1)^2-2) = ((n(n+1)-2)^2-2)
which neatly shows that the product of two adjacent numbers of the form
k^2-2 is also of that form.
So I looked at k^2-3, and to my surprise
(n^2-3)((n+1)^2-3) = ((n(n+1)-3)^2-3)
held as well, and I immediately generalized to
(n^2+k)((n+1)^2+k) = ((n(n+1)+k)^2+k)
This was a very pleasant result. The product of two adjacent numbers of
the form n^2+k is again of that form. The subexpression n(n+1)+k is a
added bonus.
I'm sure I'm not the first discoverer, I imagine this is a minor footnote in
quadratic field theory or something, but it kept me amused in the darkness.
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