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I am doing a large data-set computation. Among those computational steps, in one step I need to do a sum with a pattern: Sum the elements with same interval.

For example, for a list from 1 to 9; With the interval being set to 3 manually. So it could be other values in different cases.

And the list would be calculated as:

1 + 4 + 7 = 12;

2 + 5 + 8 = 15;

3 + 6 + 9 = 18;

So for list = Range[1,9],the final desired result would be {12,15,18} in this example. I attached an illustration for a further elaboration: sum the element with the same color when interval = 3:

Thanks for @Chris's Answer, the above case could be solved by:

 Total[Partition[Range[9], 3]]

Edit my original question from here:

But what I actually want to do is only sum "N" numbers in a time. N is settled and when N = 3 in below example:

It should be computed by :

1 + 4 + 7 = 12;

2 + 5 + 8 = 15;

3 + 6 + 9 = 18;

10 + 13 + 16 = 39;

11 + 14 + 17 = 42;

12 + 15 + 18 = 45;

Hereby the result would be {12,15,18,39,42,45}

I think this might be not hard, but I just can't think it very clearly when I want to utilize the parallelization characteristics of MMA and trying to avoid Unpacked Array results.

Total[Partition[Range[9], 3]]

{12, 15, 18}

Update for revised question:

r = Range[18]    



Total /@ Flatten[Partition[#, 3] & /@ {r[[1 ;; ;; 3]], r[[2 ;; ;; 3]], r[[3 ;; ;; 3]]}, 1]

Using the six-argument form of Partition:

Join @@ Partition[Partition[Range[9], 3], 3, 3, {1, 1}, {}, Plus]

{12, 15, 18}

Join @@ Partition[Partition[Range[18], 3], 3, 3, {1, 1}, {}, Plus]

{12, 15, 18, 39, 42, 45}

More generally,

ClearAll[partsums]

partsums[lst_List, n_Integer] := Join@@Partition[Partition[lst, n], n, n, {1,1}, {}, Plus]

Examples:

partsums[Range[18], 3]

{12, 15, 18, 39, 42, 45}

Grid[Prepend[Table[{i, Column[i Range[7]], Column[partsums[Range@#, i] & /@ 

    (i Range[7])]}, {i, {3, 4, 5}}], {"n", "Length@list" , "f[list, n]"}], 

    Alignment -> Center, Dividers -> All] // TeXForm

$smallbegin{array}{|c|c|c|}
hline
text{n} & text{Length@list} & text{f[list, n]} \
hline
3 &
begin{array}{l}
3 \
6 \
9 \
12 \
15 \
18 \
21 \
end{array}
&
begin{array}{l}
{1,2,3} \
{5,7,9} \
{12,15,18} \
{12,15,18,10,11,12} \
{12,15,18,23,25,27} \
{12,15,18,39,42,45} \
{12,15,18,39,42,45,19,20,21} \
end{array}
\
hline
4 &
begin{array}{l}
4 \
8 \
12 \
16 \
20 \
24 \
28 \
end{array}
&
begin{array}{l}
{1,2,3,4} \
{6,8,10,12} \
{15,18,21,24} \
{28,32,36,40} \
{28,32,36,40,17,18,19,20} \
{28,32,36,40,38,40,42,44} \
{28,32,36,40,63,66,69,72} \
end{array}
\
hline
5 &
begin{array}{l}
5 \
10 \
15 \
20 \
25 \
30 \
35 \
end{array}
&
begin{array}{l}
{1,2,3,4,5} \
{7,9,11,13,15} \
{18,21,24,27,30} \
{34,38,42,46,50} \
{55,60,65,70,75} \
{55,60,65,70,75,26,27,28,29,30} \
{55,60,65,70,75,57,59,61,63,65} \
end{array}
\
hline
end{array}$

Total@Take[Range@9, {#, -1, 3}] & /@ Range@3    

{12, 15, 18}

or..

Total /@ Transpose@Partition[Range@9, 3]   

{12, 15, 18}