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From: richard@cogsci.ed.ac.uk (Richard Tobin)
Newsgroups: rec.puzzles
Subject: Re: Add the numbers in a 9x9 multiplication Table
Date: Mon, 9 Jun 2025 16:32:37 -0000 (UTC)
Organization: Language Technology Group, University of Edinburgh
Message-ID: <10272b5$re5o$1@artemis.inf.ed.ac.uk>
References: <0b09b939cdbbe465809a7ec27e30912a@www.novabbs.com> <101tbqu$1rh7t$1@dont-email.me> <101v19u$2bho4$1@dont-email.me> <1026gbt$gsd0$1@dont-email.me>
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Originator: richard@cogsci.ed.ac.uk (Richard Tobin)

In article <1026gbt$gsd0$1@dont-email.me>,
David Entwistle  <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

>Given the sequence 0, 1, 9, 36, 100, 225, 441... it is possible to 
>calculate the polynomial expression for the sum of the entries in a 
>multiplication table of n rows and n columns. 2025 is the 9th entry in 
>this sequence as it is the sum for the entries in a 9 x 9 multiplication 
>table.
>
>Can you calculate that function?

   sum(x=1..n) sum(y=1..n) [xy]
=  sum(x=1..n) [x sum(y=1..n) y]
=  [sum(x=1..n) x] [sum(y=1..n) y]
=  [sum(x=1..n)]^2

sum(x=1..n) is well known and easily seen geometrically.

-- Richard