Wednesday, September 30, 2009

On the escalator

I have regular meetings in the Bahen center at U of T, which is where most of their communications people are located. However, the math department is located on the 6th floor, and if I'm early for a meeting, I usually hang out in their library.

On one recent visit to the math library, the Fibonacci Quarterly caught my eye. The lead-off article was on escalator numbers. Unfortunately I can't find an online version of the article, but another article by the same author (Grundman) is here.

A note on notation: in the text, take a[i] to mean a with subscript i. An escalator sequence a[1], a[2], ... has the property that, for all n,

(Eq. 1)

The escalator numbers A[i] are the partial sums (or products) of the escalator sequence:

Escalator sequences are defined by their starting element a[1], which cannot be equal to 1. It is easy to show that

(Eq. 2)


(Eq. 3)

The escalator number relation (Eq. 1) reminds me a lot of the arithmetic-geometric mean inequality (AGMI): for positive numbers x[1], x[2], ...,

From (Eq. 2) and (Eq. 3), we see that if A[i-1] > 1, then a[i] > 1 and A[i] > 1. Since A[1] = a[1], we can use an inductive argument to show that if a[1] > 1, then a[j] > 1 for all j, and therefore a[j] is positive for all j. Thus, if a[1] > 1, the AGMI can be applied to the escalator sequence.

After playing around a bit, I got the following (hopefully nontrivial) result. Using the AGMI, if a[1] > 1, we can show that

(Eq. 4)

This is shown as follows. First, we have

where the middle inequality is the AGMI. The rest follows from algebraic manipulation.

The bound in (Eq. 4) is surprisingly tight for large n when a[1] is reasonable. For instance, if a[1]=2, then A[100] = 106.4308..., while the bound gives 104.7616... .

Three questions. First, is this result already known? Second, is the bound asymptotically tight for large n? Third, can we get an upper bound?

The engineer in me can't help trying to think about applications for escalator numbers, but I haven't thought of anything yet.

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