One way to continue the sequence would be:
4 = 256
5 = 1024
6 = 4096
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I recall we had a teaser some while back that had a sequence 1, 2, 4, 8, 16, 31 (not 32) or something like that. It was the number of areas created by repeatedly joining lines for adding an extra point on a circle.
I can’t think of anything equivalent for powers of 4 but here’s an artificial alternative to @Mulberry’s very sensible answer:
a(n) = 4a(n-1) - a(n-3), where a(1) = 4 and a(n) = 0 for n <= 0
a(2) = 4x4 - 0 = 16
a(3) = 4x16 - 0 = 64
a(4) = 4x64 - 4 = 252
a(5) = 4x252 - 16 = 992
a(6) = 4x992 - 64 = 3904
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Well done Mulberry.
As Ravvie said, a very sensible solution. It’s the solution that I came up with when I tried this teaser years ago, and it was also the solution that the quiz-book or newspaper (I don’t recall where it came from) provided.
Cheers
Don
Hi Ravvie, I’m intrigued by your alternative answer. However, I can’t figure out what ‘a’ and ‘n’ represent in your basic equation.
For example, does ‘a’ represent the underlying digit ie ‘4’ in the teaser that I set. And does ‘n’ represent the increasing power ie 1, 2, 3, 4, 5, etc as per the teaser ? I tried this and it didn’t appear to work 
Could I substitute say 3 or 5 in your formula instead of 4 and generate the first few terms of 3^n and 5^n
Cheers
Don
“a” is a recursive function, “n” is just a label for the nth recursion. I’m limited in how I typeset on this phone.
In words the function is “the nth result is 4 times the previous result less the 3rd previous”.
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Ah! Gotcha !!
Neat.
Next time, rather than just three entries, i’ll set the sequence up with the first five entries, 1, 4, 16, 64 and 256
Cheers
Don
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Accurate Pi Day
Using the date 18th May, create an estimate of π as accurately as you can.
Whilst I have an answer in mind, the question is so open-ended that anything goes.
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Well Ravvie, I have my first approximation, but i’m not seeking any gold medals yet …
Since ‘3’ is the closest integer to Pi, and not too bad an estimate, how about …
(8 - 5) x 1 = 3 = Pi (approximately !
Now i’m sure we can all improve on that, but are the rules based on 18th May written as 18/05 or 18/5 or 18/05/2022 (or some other interpretation of 18th May ?
And do we have to use ALL the digits, and each one only once ?
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That’s a good start!
There aren’t any rules really, any liberal interpretation would do just fine. A bit like how Paul Merton interprets the rules in Radio 4’s “Just a Minute”, if you’ve ever listened to that show. They are just guidelines.
My first offering was based on a simple, single use of all three digits in 18/5 to represent 18th May.
Another, slightly better approximation, based on a similar, simple use of 18/05 would be …
(8 - 5) + (0.1) = 3.1
If however, we can use some or all of the digits in 18/05/2022 then I would initially offer the schoolboy version …
22/(8 - 1) = 22/7
Which is at least a recognised and acceptable approximation.
I’ve just realised that 18th May this year was National Maths Day (or some such designation ?)
But it was also designated to mark quite a few other events, so not entirely unique to Maths.
A quick Google reveals …
Wed May 18th, 2022 is National Numeracy Day , International Museum Day, National Visit Your Relatives Day, National No Dirty Dishes Day and a host of other Days including Women in Maritime Day etc etc
That’s some great lateral thinking! I like it.
Your other suggested answers are developing nicely too.
The sqrt(10) = 3.16278 5dp which isn’t too bad an approximation
But if we subtract 0.0212 we get 3.14158 which is pretty close to Pi at 3.14159 5dp
Hence this offering …
sqrt(5x2) - (0.0212) = 3.14158
Uses all the digits in 18/05/2022 except 8 (I could have gone sqrt(8+2) and thus left 5 out of the solution)
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Could you include the 8 on the end of 212 to get 3.131588 which rounds off to 3.14159? (Edit: just spotted the minus sign so that won’t help).
Your answer is more accurate than the one I was thinking of when I set the teaser. I will have another try to see if I can improve on your answer. You have set the bar quite high.
That possibility had crossed my mind at some stage. And was rejected for the same reason !
Young Mr R raised a very good question: “Why 18th May in particular?” but he didn’t go on to answer it. Don explored the date using Google. I wasn’t aware of National Numeracy Day so I take that to be a spooky coincidence.
However, Googling “Pi Day” reveals that it is officially a day in March, written as 3.14 in US date format. The estimate 3.14 is a little low.
There is also a Pi Approximation Day in July, written as 22/7 in a non-US date format. The estimate is a little high. I chose 18th May as it is exactly half way between the two dates.
Hence my answer is half way between 3.14 and 22/7, giving 3.1414.
Mrs R rolled her eyes at my answer, it’s not proper maths apparently.
The teaser is still open to other contributions, whether or not they surpass the above or Don’s much better estimate.
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Here’s my second contribution, using just the info in the question, with 18th May exactly as formatted with all characters used once. I have used gridded paper to avoid the clutter of loads of zeroes. Also I converted each letter to its position in the alphabet.
It is accurate enough to measure the length of the equator to within about 1 foot.
Mrs R, a university maths lecturer, remains unimpressed.
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I can understand why Mrs R is unimpressed.
As you stated a few days ago, there aren’t any rules. I’m sure that if you added 22nd July into the equation, and permitted a translation into French, German Spanish … you get the idea ? you could probably derive quite a few more decimal places …
… and still fail to impress her.