Problem:
Score : 600 points

Problem Statement

We have an integer sequence A of length N, where A_1 = X, A_{i+1} = A_i + D (1 \leq i < N ) holds.

Takahashi will take some (possibly all or none) of the elements in this sequence, and Aoki will take all of the others.

Let S and T be the sum of the numbers taken by Takahashi and Aoki, respectively. How many possible values of S - T are there?

Constraints

-10^8 \leq X, D \leq 10^8

1 \leq N \leq 2 \times 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X D

Output

Print the number of possible values of S - T.

Sample Input 1

3 4 2

Sample Output 1

8

A is (4, 6, 8).

There are eight ways for (Takahashi, Aoki) to take the elements: ((), (4, 6, 8)), ((4), (6, 8)), ((6), (4, 8)), ((8), (4, 6))), ((4, 6), (8))), ((4, 8), (6))), ((6, 8), (4))), and ((4, 6, 8), ()).

The values of S - T in these ways are -18, -10, -6, -2, 2, 6, 10, and 18, respectively, so there are eight possible values of S - T.

Sample Input 2

2 3 -3

Sample Output 2

2

A is (3, 0). There are two possible values of S - T: -3 and 3.

Sample Input 3

100 14 20

Sample Output 3

49805

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(defun main ()
  (declare #.OPT
           (inline sb-impl::stable-sort-list))
  (let* ((n (read))
         (x (read))
         (d (read))
         (table (make-hash-table :test #'eq))
         (res 0))
    (declare (uint31 n)
             (int32 x d)
             (uint62 res))
    (when (zerop d)
      (if (zerop x)
          (println 1)
          (println (+ n 1)))
      (return-from main))
    (when (< d 0)
      (setq d (- d)
            x (- x)))
    (loop for p from 0 to n
          for rem = (mod (* p x) d)
          for init-idx of-type uint62 = (ash (* p (- p 1)) -1)
          for end-idx of-type uint62 = (+ 1
                                          (- (ash (* n (- n 1)) -1)
                                             (ash (* (- n p) (- n p 1)) -1)))
          do (push (cons (+ (* p x) (* d init-idx)) 1)
                   (gethash rem table))
             (push (cons (+ (* p x) (* d end-idx)) -1)
                   (gethash rem table)))
    (labels ((recur (list prev depth)
               (declare (fixnum prev depth))
               (when list
                 (destructuring-bind (pos . dir) (car list)
                   (declare (fixnum pos dir))
                   (if (= dir 1)
                       (if (= depth 0)
                           (recur (cdr list) pos 1)
                           (recur (cdr list) prev (+ depth 1)))
                       (if (= depth 1)
                           (progn
                             (incf res (floor (- pos prev) d))
                             (recur (cdr list) 0 0))
                           (recur (cdr list) prev (- depth 1))))))))
      (loop for key being each hash-key of table
            do (setf (gethash key table)
                     (sb-impl::stable-sort-list
                      (gethash key table)
                      (lambda (x y)
                        (declare ((cons fixnum fixnum) x y))
                        (or (< (car x) (car y))
                            (and (= (car x) (car y))
                                 (< (cdr x) (cdr y)))))
                      #'identity)))
      (loop for nodes being each hash-value of table
            do (recur nodes 0 0))
      (println res)
      table)))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "C:/msys64/usr/bin/cat.exe" '("/dev/clipboard") :output out)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3 4 2
"
    "8
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "2 3 -3
"
    "2
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "100 14 20
"
    "49805
")))