What is the pattern behind these numbers?

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Discussion Overview

The discussion revolves around identifying a pattern in a sequence of numbers: 0, 3, 15, 42, 90, ... Participants explore various mathematical formulas and approaches to derive the next number in the series, as well as the underlying structure of the sequence.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that the next number in the sequence is 165, based on a formula involving T(n), the sum of the first n digits.
  • One participant presents a formula: (2 * n + 1) * T(n), while another suggests an alternative recursive formula for T(n).
  • There is a discussion about a potential typo in the recursive formula, with some participants questioning its validity for values of n greater than 1.
  • Another participant introduces a modified formula for T(n) that only applies when n is a perfect square, leading to further confusion and requests for clarification.
  • Several participants express differing views on the clarity and correctness of the formulas presented, with some suggesting simpler expressions for the nth term of the sequence.
  • One participant notes a pattern involving sums of consecutive numbers leading to perfect squares, which they find interesting but complex to articulate.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct formula or approach to derive the next number in the sequence. Multiple competing views and interpretations of the formulas remain unresolved.

Contextual Notes

There are limitations in the clarity of the recursive formulas presented, with some participants pointing out potential errors and inconsistencies in calculations. The discussion includes various assumptions about the definitions and applications of the formulas.

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0, 3, 15, 42, 90, ...

What number comes next?

There's something interesting about these numbers (at least to me). <- that's not a clue its just that this isn't just some random series, there's more to it that is mathematical, not personal or a bus route...
 
Last edited:
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i guess the next number is... 165
 
I get 165 too. The formula is (2 * n + 1) * T(n), where T(n) is the sum of the first n digits, n = 0, 1, 2, ...
 
jimmysnyder said:
I get 165 too. The formula is (2 * n + 1) * T(n), where T(n) is the sum of the first n digits, n = 0, 1, 2, ...
That works too, the formula I was using is T(0) = 0, T(n) = (T(n-1))/3 + n) * 3. The interesting part was that if you add consecutive numbers, they add to the next set of consecutive numbers, starting with a perfect square and going to the number before the next perfect square. Like 1+2 = 3 and 4+5+6 = 7+8 and so on in a pyramid sort of way.
 
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nnnnnnnn said:
T(0) = 0, T(n) = (T(n-1))/3 + n) * 3.
Is there a typo in this? It doesn't seem to work. For instance

T(1) = (T(0)/3) + 1) * 3 = (0/3 + 1) * 3 = 1 * 3 = 3 is OK, but
T(2) = (T(1)/3) + 2) * 3 = (3/3 + 2) * 3 = 3 * 3 = 9
 
jimmysnyder said:
Is there a typo in this? It doesn't seem to work. For instance

T(1) = (T(0)/3) + 1) * 3 = (0/3 + 1) * 3 = 1 * 3 = 3 is OK, but
T(2) = (T(1)/3) + 2) * 3 = (3/3 + 2) * 3 = 3 * 3 = 9

It is a typo, it works when n is a perfect square.
 
nnnnnnnn said:
It is a typo, it works when n is a perfect square.
You lost me there. Can you show me an example of it working? It seems to get the wrong answer for all n greater than 1, including perfect squares.
 
I miss stated it, it's: T(0) = 0, T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3, when n^(1/2) is an integer.

so for n = 4, its T(4) = ( T(1)/3 + 4 )*3 = (3/3 + 4)*3 = 15


It can also be n + (n + 1) + ... + (n + n^(1/2)) = (n + n^(1/2) + 1) + ... + ( n + 2*n^(1/2) )
 
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nnnnnnnn said:
I miss stated it, it's: T(0) = 0, T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3, when n^(1/2) is an integer.
so for n = 4, its T(4) = ( T(1)/3 + 4 )*3 = (3/3 + 4)*3 = 15
It can also be n + (n + 1) + ... + (n + n^(1/2)) = (n + n^(1/2) + 1) + ... + ( n + 2*n^(1/2) )
I still don't get it. In your sequence, T(4) is 90, not 15.
 
  • #10
i think it would be simpler to express the formula like this:

nth term = (2*n^3 + 3*n^2 + n)/2 , where n=0,1,2,3,4,...
 
  • #11
... or it can be this way also...

nth term = (2*n^3 - 3*n^2 + n)/2 , where n=1,2,3,4,5,... :cool:
 
  • #12
jimmysnyder said:
I still don't get it. In your sequence, T(4) is 90, not 15.

It works if you do it for n= 0^2, 1^2, 2^2, ...

then 2^2 = 15, but 4^2 = 90.
 
  • #13
croxbearer said:
i think it would be simpler to express the formula like this:
nth term = (2*n^3 + 3*n^2 + n)/2 , where n=0,1,2,3,4,...

I think that would make it simpler too. Before I figured out the recursive function, I was using excel with the formula n + (n+1) + ... + (n+n^(1/2)), that was a pain because you can't just copy and paste...
 
  • #14
nnnnnnnn said:
It works if you do it for n= 0^2, 1^2, 2^2, ...
then 2^2 = 15, but 4^2 = 90.
Work it out for me for 3^2 and maybe I'll finally get it.
 
  • #15
T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3

T(0) = 0
T(1) = 3
T(4) = 15

T(3^2) = ( T(2^2) )/3 + 3^2) * 3 = (15/3 + 9) * 3 = 42

I think the problem was that I knew what I was thinking but then I didn't clearly state it. I was intending that T(n-1) would be the next lowest perfect square, so if n=9, I had n-1=4, even though I didnt state that until later. The most recent post should be an accurate formula.
 
  • #16
Thanks nnnnnnnn, I finally get it. You could simplify your own formula greatly as follows:

G(0) = 0
G(n) = G(n-1) + 3n^2, n = 1, 2, 3, ...

Essentially this is your formula with the factor of 3 distributed and n replaced with its square root.

For example:
G(0) = 0
G(1) = G(0) + 3 * 1 * 1 = 0 + 3 = 3
G(2) = G(1) + 3 * 2 * 2 = 3 + 12 = 15
G(3) = G(2) + 3 * 3 * 3 = 15 * 27 = 42
G(4) = G(3) + 3 * 4 * 4 = 42 + 48 = 90
etc.
 
  • #17
I saw this too, and it would have been much easier to use. When I noticed the pattern (1+2=3, 4+5+6=7+8...), the perfect squares stuck out and then when I came up with the recursive function I was looking at it more from a point of view of the squares than I was from the pov of a sequense. Anyways I decided to keep it that way (even though it is more confusing and harder for me to state what I meant) because it seemed more related to the pattern.
 

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