What's The Next Number In The Sequence?

[MarkFL]

I thought it might be fun if we have a thread where a user posts the first terms in a sequence, just enough terms to that a rule for determining the next and following number can be inferred. Those responding should not only state the next number, but the rule they used to find it. Then the person who found the rule and stated the next number can give their own sequence.

I'll begin with one I "discovered" several years ago when studying rational root approximations (I later found out it is a well-known sequence).

0, 1, 2, 5, 12, ...

 

[Rens]

29
12+12+5

 

[MoreCoffee]

29
12+12+5

looks right except I suspect the formula is 2*(12) + 5 a Pell number.

 

[MarkFL]

29
12+12+5

You (and Phil) are correct. :thumbsup:

 

[Rens]

0,1,3,7,13,...
MarkFL and MoreCoffee will never find out this one.
:smirk:

 

[MarkFL]

0,1,3,7,13,...
MarkFL and MoreCoffee will never find out this one.
:smirk:

This one took some thinking!

This appears to be the number of squares of any size in a staircase of n steps built with unit squares.

For a staircase having 0 steps, there will be 0 squares.

For a staircase having 1 step, there will be 1 square:

 __
|__|

For a staircase having 2 steps, there will be 3 unit squares:

 __
|__|__
|__|__|

For a staircase having 3 steps, there will be 6 unit squares and 1 square whose sides are 2 units for a total of 7 squares:

 __
|__|__
|__|__|__
|__|__|__|

For a staircase having 4 steps, there will be 10 unit squares and 3 squares whose sides are 2 units for a total of 13 squares. And so the next number in the sequence will come from a staircase having 5 steps in which there are 15 unit squares and 6 squares whose sides are 2 units. and 1 square whose side lengths are 3, for a total of 22 squares.

 

[Rens]

This one took some thinking!

This appears to be the number of squares of any size in a staircase of n steps built with unit squares.

For a staircase having 0 steps, there will be 0 squares.

For a staircase having 1 step, there will be 1 square:

 __
|__|

For a staircase having 2 steps, there will be 3 unit squares:

 __
|__|__
|__|__|

For a staircase having 3 steps, there will be 6 unit squares and 1 square whose sides are 2 units for a total of 7 squares:

 __
|__|__
|__|__|__
|__|__|__|

For a staircase having 4 steps, there will be 10 unit squares and 3 squares whose sides are 2 units for a total of 13 squares. And so the next number in the sequence will come from a staircase having 5 steps in which there are 15 unit squares and 6 squares whose sides are 2 units. and 1 square whose side lengths are 3, for a total of 22 squares.

Huh? LOL
It's almost good.

 

[MarkFL]

Huh? LOL
It's almost good.

I'm not sure if you are saying that's the number you had in mind, but I gave a rule that fits all of the given numbers, and gave the next number in the sequence using that rule, so...ummm...here's a fun one:

1, 11, 21, 1211, 111221, ...

 

[Rens]

I'm not sure if you are saying that's the number you had in mind, but I gave a rule that fits all of the given numbers, and gave the next number in the sequence using that rule, so...ummm...here's a fun one:

1, 11, 21, 1211, 111221, ...

No it was almost good, close, it was 21.
0*0+1=1
1*1+2=3
2*2+3=7
3*3+4=13
4*4+5=21

an educated guess:
11221112
because that's a nice number

 

[MarkFL]

No it was almost good, close, it was 21.
0*0+1=1
1*1+2=3
2*2+3=7
3*3+4=13
4*4+5=21

Zero is the first number you gave and:

(-1)² + (-1 + 1) ≠ 0

an educated guess:
11221112
because that's a nice number

No, that's not it.

 

[Ackbach]

312211.

It's the number of numbers you say when you read out one digit at a time. You start with 1, then the next number is 11, meaning that I had to write "one one" to get the previous number. The next number, 21, means that in order to say the 11, I had to say "two ones". Then you had to say one two, one one. And so one.

Here's mine, and I'm changing the rules just a hair for my post. This is a finite, complete sequence. What's the rule? 8, 5, 4, 9, 1, 7, 6, 10, 3, 2.

 

[MarkFL]

312211.

It's the number of numbers you say when you read out one digit at a time. You start with 1, then the next number is 11, meaning that I had to write "one one" to get the previous number. The next number, 21, means that in order to say the 11, I had to say "two ones". Then you had to say one two, one one. And so one.

Correct! :thumbsup:

Here's mine, and I'm changing the rules just a hair for my post. This is a finite, complete sequence. What's the rule? 8, 5, 4, 9, 1, 7, 6, 10, 3, 2.

They are the natural numbers 1-10 sorted alphabetically by their English spellings?

 

[Ackbach]

They are the natural numbers 1-10 sorted alphabetically by their English spellings?

Yep!

 

[MarkFL]

Okay, here's one:

1, 2, 8, 33, ...

 

[Ackbach]

Hmm. Well, this could be:

OEIS Sequence A099015, rule Fib(n+1)*(2Fib^2+FibFib(n-1)+Fib(n-1)^2), with the next number 140, or
OEIS Sequence A172448, with next number 344, or
several other sequences.

Maybe it's Sequence A053817?

 

[MarkFL]

...Maybe it's Sequence A053817?

Yes, that's the rule I was using... I didn't realize so many other rules fit...I should have given more numbers.

 

[Ackbach]

Mark, why don't you propose another? I'll give up my turn to generate a sequence.

 

[MarkFL]

Mark, why don't you propose another? I'll give up my turn to generate a sequence.

Okay, here's one:

2, 7, 20, 54, 148, ...

 

[MoreCoffee]

Okay, here's one:

2, 7, 20, 54, 148, ...

Not sure I can work that one out ...

 

[Rens]

Not sure I can work that one out ...

tsssss easy, just take OEIS Sequence A099015, rule Fib(n+1)*(2Fib^2+FibFib(n-1)+Fib(n-1)^2)
hahahahahahahahahahaha
What on earth is this thread about?