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The parts of the function definition

The preceding analysis gives us the bones of our function definition: first, we will need a variable that we can call total that will be the total number of pebbles. This will be the value returned by the function.

Second, we know that the function will require an argument: this argument will be the total number of rows in the triangle. It can be called number-of-rows.

Finally, we need a variable to use as a counter. We could call this variable counter, but a better name is row-number. That is because what the counter does is count rows, and a program should be written to be as understandable as possible.

When the Lisp interpreter first starts evaluating the expressions in the function, the value of total should be set to zero, since we have not added anything to it. Then the function should add the number of pebbles in the first row to the total, and then add the number of pebbles in the second to the total, and then add the number of pebbles in the third row to the total, and so on, until there are no more rows left to add.

Both total and row-number are used only inside the function, so they can be declared as local variables with let and given initial values. Clearly, the initial value for total should be 0. The initial value of row-number should be 1, since we start with the first row. This means that the let statement will look like this: 

  (let ((total 0)
        (row-number 1))
    body...)

After the internal variables are declared and bound to their initial values, we can begin the while loop. The expression that serves as the test should return a value of t for true so long as the row-number is less than or equal to the number-of-rows. (If the expression tests true only so long as the row number is less than the number of rows in the triangle, the last row will never be added to the total; hence the row number has to be either less than or equal to the number of rows.)

Lisp provides the <= function that returns true if the value of its first argument is less than or equal to the value of its second argument and false otherwise. So the expression that the while will evaluate as its test should look like this: 

(<= row-number number-of-rows)

The total number of pebbles can be found by repeatedly adding the number of pebbles in a row to the total already found. Since the number of pebbles in the row is equal to the row number, the total can be found by adding the row number to the total. (Clearly, in a more complex situation, the number of pebbles in the row might be related to the row number in a more complicated way; if this were the case, the row number would be replaced by the appropriate expression.) 

(setq total (+ total row-number))

What this does is set the new value of total to be equal to the sum of adding the number of pebbles in the row to the previous total.

After setting the value of total, the conditions need to be established for the next repetition of the loop, if there is one. This is done by incrementing the value of the row-number variable, which serves as a counter. After the row-number variable has been incremented, the true-or-false-test at the beginning of the while loop tests whether its value is still less than or equal to the value of the number-of-rows and if it is, adds the new value of the row-number variable to the total of the previous repetition of the loop.

The built-in Emacs Lisp function 1+ adds 1 to a number, so the row-number variable can be incremented with this expression: 

(setq row-number (1+ row-number))