9,73,241,561,1081,1849,_?_

<a href="http://www.wolframalpha.com/input/?i=9%2C73%2C241%2C561%2C1081%2C1849" rel="nofollow">http://www.wolframalpha.com/input/?i=9%2C73%2C241%2C561%2C10...</a>

The wonderful (satirical, but mathematically sophisticated) book Mathematics Made Difficult introduces difference tables, and uses them to demonstrate how to compute the next number in the sequence [1,2,4,8,16,…] – that number being, of course, 31.<p>This answer is correct in that it is the next item in the lowest-order polynomial that generates the first five terms. This reveals both the strength and weakness of difference tables (and the flavor of Mathematics Made Difficult).

I read the title here which has no spaces between the numbers (a real pet peeve :P) then went on to read the explanation of "Edwin's" attempts to solve the blank. Except without the spaces, I thought he was trying to find the remaining digits of a single number (with commas for thousands separator).<p>(before clicking the link, I was expecting "find the remaining digits to make this number a prime" or something)<p>Nice link though!

I discovered difference tables as a kid, didn't know they were used much, I was modelling motion on a 2D display (an oscilloscope hooked to a P2P11!). Was a lifesaver for quick polynomial evolution - uses only addition, execution time scales linearly with the order of the polynomial. So, how do you go from the difference table "coefficients" to the polynomial?

I hit my head to wall after reading this. Difference table is a really cool way to solve this kind of questions. I thought "how come I have never learned this difference table before". I could make better points with IQ tests with difference tables :)

Difference table only works in case of arithmetico-geometric series or polynomial series. It fails in other sequences like fibonacci,prime etc. However if given a sequence I always try difference table first as there are high chances that it'll be cracked.

I learned about difference tables in Conway and Guy's great book "The Book of Numbers". I wish the book had been available when I was an undergrad. It describes everything I loved in math that I wasn't able to pinpoint. There are so many little gems in it, I've spent months reading it halfway through and then starting over to pick up what I missed. Highly recommended. <a href="http://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X" rel="nofollow">http://www.amazon.com/Book-Numbers-John-H-Conway/dp/03879799...</a>

This question is not well-defined. Any number can be a solution.

Not to be too harsh here, but...<p>I've been using "difference tables" (without calling it that) since I was 10 years old. I don't mean this to be bragging at all, because I didn't think (and still don't think) it was at all remarkable. It's just a basic method of analysis.

<a href="http://www2.research.att.com/~njas/sequences/" rel="nofollow">http://www2.research.att.com/~njas/sequences/</a><p>This is a great resource for finding information on integer sequences significant in combinatorics and other formal math topics. It didn't have the solution to this particular sequence, because, afaik there is nothing particularly interesting about it.

You can see a discussion about difference tables here: <a href="http://www.physicsforums.com/showthread.php?t=114995" rel="nofollow">http://www.physicsforums.com/showthread.php?t=114995</a>

Was surprised to see this happen:<p><a href="http://www.wolframalpha.com/input/?i=0%2C1%2C2%2C4%2C8%2C16%2C32%2C64%2C128%2C256%2C512" rel="nofollow">http://www.wolframalpha.com/input/?i=0%2C1%2C2%2C4%2C8%2C16%...</a>

I loaded into excel, then copy and dragged down to see what excel would say is the next 2 answers ... I got the following<p>...1081,1849,[1890.067],[2248.467].... Now I just wonder how excel got these answers :/

This method of computing successive terms of a polynomial actually has an interesting place in the history of computers; the basic principle is the same as that used by Charles Babbage's difference engine.<p>Wikipedia article on the difference engine: <a href="http://en.wikipedia.org/wiki/Difference_engine" rel="nofollow">http://en.wikipedia.org/wiki/Difference_engine</a><p>Just for fun, a difference engine built with Legos: <a href="http://acarol.woz.org/" rel="nofollow">http://acarol.woz.org/</a>

The series 9,73,241,561,..corresponds to:<p>2^3+1^2; 4^3+3^2; 6^3+5^2; 8^3+7^2; 10^3+9^2; 12^3+11^2; ?<p>Soln: 14^3+13^2 = 2913

If somebody wants to try this method on different sequences, I made a simple solver in JavaScript:<p><a href="http://alteredqualia.com/visualization/hn/sequence/" rel="nofollow">http://alteredqualia.com/visualization/hn/sequence/</a>

Is there interest in people posting neat math puzzles on HN more frequently? There's a few I really like and I would love to see what puzzles others have.

oleis turns up nothing, sadly

Can anyone explain why it takes so long to find the equation differentially? I understand that is subjective, but the 'constants' are only constant from y''' to y''. From y'' to y' the 'constant' is linear (meaning you have to solve for change in y' - y) and the 'constant' is quadratic from y' to y.<p>Basically you can't just integrate each DE and solve for a constant C by finding the value of the lower order DE where x=1 and finding the difference. Why not?<p>I'm in Calc BC in high school so this could be a stupid question.

why is this at the top?

Please don't post your homework to the list.