15, 34, 65, 111, 175, ... Bonus points if you can explain this patterns significance.

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I believe that the next two numbers are 260 & 369.

More of the series might be: 1, 5, 15, 34, 56, 111, 175, 260, 369, 505 . . .

The equation that generates the series is

M = 1/2*2*(n^2 + 1)

This is the series of magic constants, that is, the sums and rows of magic squares where the square starts with a 1. A 3 x 3 magic square containing the numbers 1 through 9 will have rows and columns totaling 15. A 4x4 magic square starting with 1 will total 34; a 5x5 will total 111; and so forth.

I had to put on my thinking cap, then search the web, then put on my thinking cap again to figure this one out! Thanks, CWanamaker!

Check out these links to learn more:

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Well, there you are now! I couldn't work out an equation for it but when I look at yours, I wouldn't have worked it out anyway! LOL. It's always nice to see that bit of extra information.

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I think the next two numbers are 260 and 369.

I put the numbers into a spreadsheet and subtracted them, to find the difference between them. It didn't seem to have any particular pattern, so then I subtracted the differences between the differences. That came out into a pattern of 1, 15, 18. So I then assumed the pattern was to add 3 to the 2nd set of differences and just worked my way back up the spreadsheet to come up with these numbers.

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I worked out the cubic relation like Meg, but I didn't know what made the numbers special, so I cheated and plugged it into OEIS

In addition to being the sequence of magic square constants, the nth term is the sum of the integers between T(n-1) + 1 and T(n) (inclusive), where T(n) is the nth triangular number. (For this relation to hold, you have to start the sequence as 0, 1, 5, 15, 34, ... with 0 as the 0th term.)

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