Here is an interesting case of numerical explorations leading to mathematical results.
Grade-eleven me was interested in the relationship between parabolas and exponentials,
and thought that the infinite series
\[S = \sum_{n=1}^\infty \frac{n^2}{2^n}\]
had something to do with it.
Aiming to sum this series,
I took the sequence of partial sums *S _{n}*
and "cleared denominators" to obtain
the integer sequence

n
| T
_{n} | ΔT
_{n} | Δ^{2}T
_{n} | Δ^{3}T
_{n} |
---|---|---|---|---|

1 | 1 | |||

5 | ||||

2 | 6 | 10 | ||

15 | 12 | |||

3 | 21 | 22 | ||

37 | 24 | |||

4 | 58 | 46 | ||

83 | 48 | |||

5 | 141 | 94 | ||

177 | 96 | |||

6 | 318 | 190 | ||

367 | ||||

7 | 685 | |||

Now, this was all just detective-work: the fact that *T _{n}* appeared to be a linear combination of a square, an odd number, two, and six times a power of two had only been shown for

*T _{n}* = 6·2

because, for example, *T*_{3} = 21 = 48 − 2 − 9 − 16 = 6·2^{3} − (2·3 + 3) − (3 + 1)^{2}.
The formula for *T _{n}* simplified to 6·2

*S _{n}* = 6 − (

This final formula is *correct*: it can be verified by induction (as it was by grade-eleven me).
(To do this,
call the right-hand side *S _{n}*′,
show that