Back of an Envelope Math

I think you nailed it!

I think I love you !

It maybe needs a slight tweak, because 1st term is 0, 2nd is 1, etc. Plug in a number and I think you'll see that 'n' need to be 'n-1' or something like that. My brain is slightly fried, so excuse me if wrong on that minor detail. It's an A +++ answer.

To satisfy the curious, TV was garbage and some earlier mention of numbers set my brain in motion. Using dot notation ("." = "x");

1 + 6.0

2 + 6.1

3 + 6.4

4 + 6.10

5 + 6.20

6 + 6.35

7 + 6.56 etc

The sequence is cube numbers. Looking at your answer, and provided tonight's TV is just as bad, I may possibly figure out why I managed to get full circle.

Sane people may wish to stop reading at this point.

Trisection attempts take up too much paper, and 'UK has Talent' (or whatever it was) only lasts so long. I've a horrible fear that 'I'm a Celeb' starts tonight, so my attempts on Fermat's margin may take the whole thing modular later. I wish I could stick a 'tongue-in-cheek' smiley in here, but... (). It can't be done, therefor TV boredom almost demands that I waste ink trying. Blame Mizuti - it's all down to his mention of the film "Pi" a few months back, and the annoying number 216

Many thanks, SolarEagle , you are most excellent. Another well deserved GA from me.

What is the significance of the line above the nimber 3 (in the power of n)? Or is something missing above the line? i.e. 1/3 power?

How is this a formula for getting the top line?

This is not really a formula as you state it well but a visual display. We used this in the 60's without microsoft. (LOL)

It is just a practical help to get the top line. How it works:

the lower serial 3.4.5.6. etc goes up with 1 digit - after 8 comes 9, 10, 11 and so it can go on.

You need to bring it to the second line by adding the last number of the 2nd line in this case as example 9 to 36 = 45

45 plus 120 will be the next number on the top line. (165)

It works when the internet is down and wiki is not available and when you have no computer or excel.

Here a little further

Works pretty good in a lot of situation, generating pythaogean triples for an eg. I like such techniques because they get to the basics. A bit more work (well, a lot more than my brain could do at the time), and you come out with a formula as good as SE's.

StandardsGuy: You asked about the line above thing. It's a new one on me but, as best I recall when looking at it that, it's a bit like factorial notation. In that instance meaning (n)x(n+1)x(n+2)...An underline would indicate descending - (n)x(n-1)x(n-2) etc. I've more than likely completely mangled that but, if you look at the expanded form (ie compare to the first equation given by SE) it should become clearer.

Much thanks to all, it really was some late night confusion doodle fun. No cats have been harmed in sating my curiosity. Strange, but asking what is essentially a basic 101 question can produce some good input (and I include clarification about my mixing up the terms sequence/series).

On the 3 with the line over it. It didn't get a satisfactory answer from Googling it. I could understand never-ending for a decimal (i.e. 0.3 with a line to mean 0.333333...) So does it mean n^3 + n^2 + n? If nobody understands it, then 8 good answers were because he posted something, but nobody verified that it works.

OK this makes more sense than your first "formula". The bottom line is a sequence of +1 (starting with 2). you add up the two numbers to get the next line and do the same for the 3rd (top) line it seems.

I could have caught up on whatever point, but I did the check on some numbers to see if there was a continuity. In this example you can skip some parts of the start sequence. The formula supplied by Solar Eagle works for the first part. The second = I need some explanation of exactly what you are asking for. The upper scored 3 and the 3! I just do not understand, but I am very dumb.

Thanks, and GA to you. I knew about factorials, but the other was new to me. I'm glad I'm not the only one in the dark about this.

Hey, I pretty much said that at #14 ! (15, 16, 17 ,18, 19......). Nah, just joking, credit where due for a clear explanation.

Just a slight 'by the by', but people from different tech backgrounds seem to go for different notation. With three unknowns, xyz is often used. ijk and pqr are also much used. The meaning is fairly clear in this context, but try getting people from different disiplines to agree on xyz in a graphical context. Each of those three has a 'usual' place in UK convention, but I'd be surprised if that is the same as usage everywhere.

It's worth a GA for adding clarity to a little known/used form of expresion.

I agree. Those are good points we should always keep in mind. We've all seen posts where someone asks a question which can only be answered by knowing where the poster is located and being familiar with the local rules and regs. Posters and respondents all need to keep in mind that someone half a world away will not necessarily understand what is being said. In my last reply I used the word university instead of college because most of the world says university where we Americans say college.

I know series and sequences because I spent a lot of time with them while working on my Math degree. At that time we did not have the Internet, but we had an Oxford English Dictionary sized book listing sequences. That info is now on the website I posted in my first reply, but much improved because the website has hyperlinks to much more information for those who want to dig deeper.

We like to think Math notation ("Math" being the common nickname in the US while most of the English speaking world says "Maths") is standardized, but I've read books, articles, and papers from various countries and various times and deciphering the notation is sometimes more trouble than trying to understand the concept being presented.

I'll look like a suck-butt, but I very much agree with what you say. It's not 'on topic' - the original post was hardly serious - but your comments are well worth reading. The rating system needs redefining. More than a few threads have posts which, whilst not to the topic, are very worth a read,

Alternatively, the expression in question could also be viewed as:

(x+(m-1))!

and, where m=3;

(x+(3-1=2))! => (x-2)*(x-1)*(x-0) => (x-2)*(x-1)*x

he said, hopefully helpfully...

x(n+2) = x(n) + n*n

only when x = n²/2

Try http://oeis.org/wiki/Welcome

To pick a nit, your first sentence is incorrect and your second sentence is correct. What you presented is a sequence, an ordered list of integers. A series is an addition problem which would be notated with a ∑ followed by the formula in the excellent answer SE provided.

If you try y(n) = (x^2 -1)*(x/6) for n=1, 2, ...

you should get:

y(n=1) = 0

y(n=2) = 1

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y(n=8) = 84

y(n=9) = 120

y(n=10) = 165

y(n=11) = 220, etc.

In any case, Excel(R) can be a big help...

In answer to the OP's request for a formula, I simply offer y=((x^2 - 1)*(x/6), for x= to 1,2,3,...

y(x=1) = ((1)^2 - 1) * ( 1 / 6 ) = ( 1 - 1) *(1/6) = 0*(1/6) = 0

y(x=2) = ((2)^2 - 1) * ( 2 / 6 ) = ( 4 - 1 )*(2/6) = 3*(2/6) = 1

y(x=3) = ((3)^2 - 1) * ( 3 / 6 ) = ( 9 - 1 )*(3/6) = 8*(3/6) = 4

and, so on...

Yes, Excel (R) was a major aid in answering this question.

Who has offered a more simple one ?...

'Who has offered a more simple one?'

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Oooohhhhh! I know the answer this one! SolarEagle and DVMDSC both offered simple correct answers and were also very expedient.

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_{Did I get that right? Was I the first? How many more do I need to claim the giant stuffed animal?}

Mr. Guest offered yet another good answer.

The whole thing is a bit circuitous because I was looking for alternate ways of expressing cubic numbers. I think I stated at the off that it was a bit of late night madness, something set me thinking about the whole x3 + y3 = z3 thing, and I was just seeing what alternate ways existed to express cube numbers.

Bit naughty of me not to be explicit, but I did say it was some late night fun with an envelope.

Fermat may rest quite merry in his grave. By way of contrition, I commend readers to look up Simon Singh's book on the topic. Singh is very worth a read on several topics.

I took a long mathematical wander down a possibly similar path. It was sparked when I noticed what I thought were some odd patterns in square triangular numbers, like that certain square triangular numbers are the square of the difference between certain smaller square triangular numbers and that the next triangular root of a square triangular number could be determined by the square roots of two of the previous square triangular numbers.

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Fermat, of course showed up again and again. 'There is not enough room in the margin' indeed, I hope he is somehow still enjoying that.

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At any rate, that wander eventually introduced to me some of the uses for Pell numbers.

Last minute thought - Several people have mentioned Excel. Long time since I've used it, but as I recall you can plot data and do a 'best fit'- curve. Does anybody know if Excel will also churn out a 'best fit' equation for any given curve ? From answers given, I suspect that's a 'yes'. I should try this, but I hate chucking control over to a computer program. If something doesn't plot linear then I start trying various log scales etc. The more obvious method is the brilliant 'as requested' one given - looking at differences between numbers. Quote marks used because there are two exceptional answers in this thread; the direct answer and the method. All contributions have been much appreciated. Thank you all.

Cubic Regression Using Minitab:

Where x is an element of {0,1,2,3,4,5,6,7,8}, y= 0.000000 + 0.3333x + 0.5000x^2 + 0.1667x^3

Where x is an element of {1,2,3,4,5,6,7,8,9}, y= - 0.000000 - 0.1667x - 0.000000x^2 + 0.1667x^3

Both with correlation coefficient of 100.0%

Standard Statistics Caution: While these equations can be safely used to predict responses using values within the input range, it is not advised to use these equations to predict responses for values outside of the input range.

'....

Where x is an element of {0,1,2,3,4,5,6,7,8}, y= 0.000000 + 0.3333x + 0.5000x^2 + 0.1667x^3

Where x is an element of {1,2,3,4,5,6,7,8,9}, y= - 0.000000 - 0.1667x - 0.000000x^2 + 0.1667x^3

Both with correlation coefficient of 100.0%

...'

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I appreciate the 'caution' that you provided, but is seems like there may be a more fundamental problem. The equations above seem to indicate something that I don't believe is true. Namely:

.

Where x is an element of {0,1,2,3,4,5,6,7,8}, 0.000000 + 0.3333x + 0.5000x^2 + 0.1667x^3= - 0.000000 - 0.1667(x+1) - 0.000000(x+1)^2 + 0.1667(x+1)^3

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Multiplying everything through by 6 for clarity .........

0.000000 + 2x + 3x^2 + x^3= - 0.000000 - (x+1) - 0.000000(x+1)^2 + (x+1)^3

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or.....; 2x + 3x^2 + x^3= - (x+1)+ (x+1)^3

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Actually on closer inspection, there is nothing wrong with that. My mistake. I'm going to post this anyway on the chance that someone else starts to go down the same path.

I included everything, the zero terms and all the decimal places, just so everyone could see what kind of output Minitab produces. I try to always give away everything I've got so that my work can be reviewed and reproduced or so that my own mistakes can be made apparent. The point is to get to the right, or at least best, answer. If I'm wrong I need to know it.

I wasn't trying to give you a hard time, the equations simply did not look equivalent (with one as 'x' and the other as 'x+1') to me, but as I wrote the message and then simplified, it became apparent that the two were indeed equivalent.

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For humility and because it seemed interesting, and to save anyone else headed that way I figured I might as well go ahead and post it.

As you say, the difference in the equations is just what number is the initial integer 0 or 1.

Minitab is a new one one me. It looks interesting, but also pay to use <gasp>. I can't really justify spending dosh on that, though I'm not knocking it in any way. The mention is possibly going to be helpful to someone.

Just in passing, I rate GeoGebra as a very good freebie for 'doodles'. BCalc is also a very handy desktop freebie for crunching big numbers.

Minitab is great for my line of work because it is tailored to Reliability, Quality, and Lean Six Sigma problems.

I wouldn't pay for it out of my own pocket. Fortunately, my employer is willing to pay for it. (So, yes, I was using my work time to provide that quick regression analysis.)