Define starting vector i <- c(1,2) and then use a loop that in each iteration adds 1 to the largest element if the vector and appends it to the vector. Use colSums or whatever makes sense for the summarization after subsetting df[,i,drop=FALSE]. An experienced user like you will for sure get this going.

Edit: Here is one way to do this with a two-liner using base R:

# example data
ncols <- 6
nrows <- 3
set.seed(1)
data <- data.frame(matrix(rnorm(ncols*nrows), ncol=ncols))
# actual code
iter <- lapply(2:ncols, function(x) 1:x)
lapply(iter, function(x) rowSums(data[,x,drop=FALSE]))

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