Sage provides `fibonacci`

to access elements of the Fibonacci sequence.

This can be combined with basic list comprehension and sum.

So, you could do things like listing or summing consecutive terms:

```
sage: [fibonacci(j) for j in (0 .. 12)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
sage: sum(fibonacci(j) for j in (0 .. 12))
376
```

or non-consecutive terms:

```
sage: [fibonacci(j) for j in (1, 2, 4, 7, 11, 16)]
[1, 1, 3, 13, 89, 987]
sage: sum(fibonacci(j) for j in (1, 2, 4, 7, 11, 16))
1094
```

It seems the indices you are interested about are of the form $(n^2 + n + 2)/2$

```
sage: [(n^2 + n + 2)//2 for n in range(10)]
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]
```

so here are the terms of the Fibonacci sequence for these indices:

```
sage: [fibonacci((n^2 + n + 2)//2) for n in range(10)]
[1, 1, 3, 13, 89, 987, 17711, 514229, 24157817, 1836311903]
```

and here is their sum them up to some point:

```
sage: sum(fibonacci((n^2 + n + 2)//2) for n in range(10))
1861002754
```

and here is how this sum evolves when you push it further and further:

```
sage: [sum(fibonacci((j^2 + j + 2)//2) for j in range(n)) for n in range(10)]
[0, 1, 2, 5, 18, 107, 1094, 18805, 533034, 24690851]
```

Since all the Fibonacci numbers are at least one, if you add infinitely many of them,
the sum will tend to infinity.