# Two index-summation loop
**URL:** https://discourse.julialang.org/t/two-index-summation-loop/86664
**Category:** New to Julia
**Tags:** question
**Created:** [September 2, 2022, 12:52am UTC](https://discourse.julialang.org/t/two-index-summation-loop/86664 "2022-09-02T00:52:02Z")
**Posts on this page:** 3
**Page:** 1
### Author:  [@Longblackcoffee](https://discourse.julialang.org/u/Longblackcoffee)
#### Post date: [September 2, 2022, 12:52am UTC](https://discourse.julialang.org/t/two-index-summation-loop/86664/1 "2022-09-02T00:52:02Z")
Hi everyone.
I am trying to program the following summation into Julia:
y\_i = c\_i + \sum\_{r=1}^{i-1} b\_r\*y\_{i-r}
So for example, in the following toy numerical example:
```julia
using DataFrames
test = DataFrame(months = 0:3, y = 0
test[!,:y] = convert.(Float64,test[!,:y])
c_test = [0,1,1.1,1.2]
b_test = [0,2,2.1,2.2]
```
the correct code would spit out the following column-vector (working out can be provided if required):
y' = [0,1,3.1,9.5,29.01]
I tried different ways of using forloops but got nowhere with them. I know the following lines are completely wrong but I don’t know where to even begin to fix it
```julia
for i in 2:nrow(test)
for r in 2:nrow(test)
test[i,2] = c_test[i,1] + sum(b_test[r,1]*test[i-r,2])
end
end
```
Any help would be really appreciated, tysm!
---
### Author:  [@jling](https://discourse.julialang.org/u/jling)
#### Post date: [September 2, 2022, 1:07am UTC](https://discourse.julialang.org/t/two-index-summation-loop/86664/2 "2022-09-02T01:07:53Z")
I think your “correct result” is wrong
```julia
julia> for i in eachindex(y)
second_term = 0.0
for r=1:i-1
second_term += b_test[r] * y[i-r]
end
y[i] = c_test[i] + second_term
end
julia> y
4-element Vector{Float64}:
0.0
1.0
1.1
3.2
```
---
### Author:  [@digital\_carver](https://discourse.julialang.org/u/digital_carver)
#### Post date: [September 2, 2022, 8:40am UTC](https://discourse.julialang.org/t/two-index-summation-loop/86664/3 "2022-09-02T08:40:21Z")
> [@Longblackcoffee](#):
>
> y\_i = c\_i + \sum\_{r=1}^{i-1} b\_r\*y\_{i-r}
It seems the expression you want is:
y\_i = c\_i + \sum\_{r=1}^{i-1} b\_r\*y\_{i-r+1}
i.e. y\_{i-r+1} instead of y\_{i-r} . Implementing that gives the result you expect.
```julia
julia> test= DataFrame(months = 0:3, y = 0.0)
4×2 DataFrame
Row │ months y
│ Int64 Float64
─────┼─────────────────
1 │ 0 0.0
2 │ 1 0.0
3 │ 2 0.0
4 │ 3 0.0
julia> for i in 1:nrow(test)
test.y[i] = c_test[i]
for r in 1:i - 1
test.y[i] += b_test[r] * test.y[i - r + 1]
end
end
julia> test
4×2 DataFrame
Row │ months y
│ Int64 Float64
─────┼─────────────────
1 │ 0 0.0
2 │ 1 1.0
3 │ 2 3.1
4 │ 3 9.5
```