Re: What is the general formula for the series 1, 2, 4, 7, 11, .........

In article <fLGdnQ64XLkdninYnZ2dnUVZ8taknZ2d@xxxxxx>,
"Nick" <tulse04-news1@xxxxxxxxxxx> wrote:

"Alfred Flaßhaar" <BueroFlasshaar@xxxxxxxxxxx> wrote in message
news:51gvdtF1k75o7U1@xxxxxxxxxxxxxxxxxxxxx
pacman18_2008 wrote:
What is the general formula for the series: 1, 2, 4, 7, 11,....???

(...)

Consider the differences between the numbers. You will find out a well
known set. Write some members of your sequence in a recursive form and
after this write some members by replacing backward. Vou will find a well
known sum. Make an assumption an proof it using the method of induction.

I have never come across or forgotten what the recursive form is. I
understand from Googling that a recursive form is as compared with a closed
form.

y(n+1)=g(y(n))

whereas a closed form can express y=f(n) directly.

Nick

In this case, a recursion formula involving the prior two members of
the sequence, a la the Fibonacci sequence, is
y(n+2) = 2*y(n+1) - y(n) + 1


Alternately, using finite differences,

y: 1 2 4 7 11...
dy: 1 2 3 4 ...
ddy: 1 1 1 ...

indicates a quadratic closed form formula, which can be fitted by trial
an error, or a number of other ways.
.

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