Numbers and Patterns

[Steven G]

It always amazes me when I learn about patterns in numbers.
I just learned that the sum of two consecutive triangular numbers is a perfect square.
Numbers are person-made and these beautiful results arrive--sometimes out of nowhere. Just amazing.
Which of you knew that T_n-1 + T_n = n²?

 

[lookagain]

It always amazes me when I learn about patterns in numbers.

Numbers are > > > person-made < < < and these beautiful results arrive--sometimes out of nowhere.

Patterns exist in numbers whether people have ever existed or not. And, as for numbers, as
Martin Gardner stated (in paraphrasing), there were "two dinosaurs" in a certain place at one time.
There need not have existed any people for that to be a fact.

 

[lev888]

Patterns exist in numbers whether people have ever existed or not. And, as for numbers, as
Martin Gardner stated (in paraphrasing), there were "two dinosaurs" in a certain place at one time.
There need not have existed any people for that to be a fact.

If a tree falls on a dinosaur, can the sound the dinosaur makes be modelled with a sine function?

 

[topsquark]

If a tree falls on a dinosaur, can the sound the dinosaur makes be modelled with a sine function?

Only if no one was there to hear it.

-Dan

 

[pka]

It always amazes me when I learn about patterns in numbers.
I just learned that the sum of two consecutive triangular numbers is a perfect square.
Numbers are person-made and these beautiful results arrive--sometimes out of nowhere. Just amazing.
Which of you knew that T_n-1 + T_n = n²?

[imath]\begin{gathered} ~~~ {T_n} + {T_{n + 1}} \\ ~~~ \dbinom{n}{2} + \dbinom{n+1}{2} \\ \dfrac{n!}{2!(n-2)!}+\dfrac{(n+1)!}{2!(n-1)!}\\ \dfrac{n(n-1)}{2}+\dfrac{(n+1)n}{2}\\ \dfrac{n^2-n}{2}+\dfrac{n^2+n}{2}\\n^2 \end{gathered} [/imath]

 

[lev888]

 

[lev888]

Another interesting pattern: