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Been staring at this for a few hours and I really need some help. The sequence is:

{4, 28, 148, 700, 3124, ...}

According to google, the formula is: $$4(4^n - 3^n)$$

But I have zero idea how to come up with this on my own.

Any help or clues on how to solve this would be appreciated.

Thank you!

2 Answers 2

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The formula would be correct if the sequence obeyed $$ x_{n+2} = 7 x_{n+1} - 12 x_n. $$ You can check that.

As far as a method, the relevant one is the "number wall" method in Conway and Guy, The Book of Numbers

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    Interesting reference. I didn't know the method. I thought (it is what I write in my answer) that you had found coefficients 7 and -12 by applying the (classical) method of characteristic equation. – 2017-01-22
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    Thank you very much for having given all the pages. btw: Sorry for having given you a Hungarian name :) – 2017-01-22
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    We do not really know where the name started, except that a male ancestor arrived in New York at about 1850, and by the time he did anything official the name was written Jagy. I have come to think it was the Swiss German name Jaggi, where the a has an umlaut. I also met a Swiss man at a conference who pronounced it for me. No way of being sure. – 2017-01-22
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    Coming hopefuly from "jäger" = "hunter" ? The history of family names is funny... – 2017-01-22
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    I also have a german origin family name (Becker=baker) though I am French... – 2017-01-22
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    Probably Jager, as you say. There is a website for Ellis Island, where European immigrants arrived at the time. They say that most names got taken down correctly. However, it was typically weeks before a person got a job, or rented a place, or children began school; revised spellings tended to appear at these later events. Oh, I used to play soccer with a guy from Metz. He is back in France now. – 2017-01-22
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    I personally come from the Metz area (Lorraine region) – 2017-01-22
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    Growing up, I was a fan of the New York Mets. I tried to get Yann a Mets baseball cap, I think I was not able to find one in time. – 2017-01-22
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    @WillJagy: Nice ref! (+1) – 2017-01-24
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Will Jagy has maybe found the recurrence $$u_{n+2}=7u_{n+1}-12u_n $$ by noting that $4$ and $3$ (which are taken to the power $n$ in your formula) are the roots of $(x-4)(x-3)=x^2-7x+12=0$ explaining (...) the coefficients $7$ and $12$ of this second order recurrence, that can hardly be found by intuition.

In fact, it is not mysterious. Here is an explanation. (See also this reference)

A linear second order recurrence relationship with constant coefficients:

$$u_{n+2}=au_{n+1}+bu_n \ \ \text{with given} \ \ u_0, u_1$$

has, for its general term, an explicit expression:

$$u_n=Ar_1^n+Br_2^n,$$

where $r_1$ and $r_2$ are solutions of the quadratic equation $r^2-ar-b=0$ (under the condition that its discriminant $\Delta \neq 0$), $A,B$ being constants obtained by considering the first values of the sequence (here $A=4$ and $B=-4$.)

(this result can be proven, e.g., by recurrence on $n$.)

Remark: This method of solution "by solving a characteristic equation" is very akin to the way one solves linear differential equations with constant coefficients like $y''-7y'+12y=0$ associated with certain initial conditions.

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    I put a few pages that are probably enough to figure out the number wall method. If you can't download them I can email them. My original answer was simply saying what the linear recurrence needed to be if the reported answer in the question were to be correct. i did not actually work out anything. – 2017-01-22