Alex Braune, general manager of the Vista Hotel, and formerly of the Hotel Imperial in Vienna, tells this story: In 1957 King Saud of Saudi Arabia flew into Vienna with 17 black Cadillacs and a large guard of honor. After checking into the Hotel Imperial, his party called down for 100 pillows. No one could imagine why. But when the hotel staff went to clean the rooms, the suites looked like the inside of a down comforter. The Saudis had been having saber and pillow fights. Even the wallpaper had to be replaced.

Heads of state are not as imperial in their demands as they used to be. Even so, Washington hotels still know how to give king-size welcomes.

Their Royal Highnesses the Grand Duke and Duchess of Luxembourg, on a state visit to the United States, yesterday were royally welcomed in the Vista Hotel's presidential suite, in a 25-room separate wing.

Awaiting them:

* A box made of chocolate decorated with a marzipan Luxembourg flag, full of Belgian and American chocolates;

* A bathtub big enough for eight;

* A room full of exotic flowers in the Luxembourg colors, red, white and blue;

* Silver trays of tea cookies and huge baskets of fruits (including kiwis);

* Givenchy plush terry-cloth robes and cosmetics;

* Stocked bar and chilled American wines to go with hot and cold canapes;

* Their own concierge who speaks several languages.

Such accommodations are not uncommon in Washington's dozen or so grand hotels, which cater to international dignitaries -- and there have been many. In the past two years or so, since Blair House, the president's official guest quarters, has been closed for repairs, pomp and circumstance have made these elegant hotels hosts to hereditary and elected potentates from every continent. The Reagan administration has had 24 full state visits, including seven this year. (A state visit is one by a head of state who has been invited for a week with full trappings, including a state dinner at the White House, as guest of the president. Presidents or kings are heads of state. Prime ministers or premiers are heads of government and don't rate a full state visit.) In addition, there have been 11 official visits, 65 official working visits, plus an uncounted number of heads of state or premiers, foreign ministers or royal relatives who came to conferences, summits or on private visits.

But of all the Washington hotels, the competition is perhaps fiercest between the Vista, opened in 1983, and the Madison, its 21-year-old neighbor.

Since the Vista's opening, the long black cars bearing visiting heads of state and other dignitaries haven't been stopping only at the Madison, as they used to do. So many have been making the circle to the Vista, on M Street between Vermont and 15th, that the Vista is adding an extra presidential suite, to be finished in time for the Inauguration.

"They stay at the Vista," said its general manager Alex Braune, waving his hand toward the hotel's 14-story atrium, "because we're new, fresh, have a different, more contemporary approach. And, of course, you have to have a feeling for grandeur and splendor."

Marshall Coyne, owner of the Madison, explains it this way: "We can't accommodate them all. The attraction of the Vista is in being close to the Madison."

In the past two years, the Madison set the record, by sheltering 24 heads of state and chiefs of government, not to mention two foreign ministers and a defense minister, according to the State Department's protocol office. The Vista in the same period has entertained 10 such potentates. But if you count only since January, the two hotels are running closer together with eight for the Madison and six for the Vista. The Four Seasons Hotel in Georgetown boasts of seven state and official visitors in the past two years, but just two this year.

Though no one wants to talk about money, the Four Seasons charges state visitors its regular suite rates -- $1,000 for the presidential suite with two bedrooms, $925 for its new royal suite with two bedrooms. Most other hotels are willing to accept the State Department's per diem, said by some to be only about $75 a day. The State Department picks up the tab for the official party, limited to 13, but Protocol Chief Selwa Roosevelt won't say just what the amount is.

"They're guests," she says, shocked that anyone would ask.

Rose Narva, president and managing director of the Hay-Adams Hotel, which in March finished a hotel-wide remodeling and redecorating, says the State Department rate is about $65 to $75 a room. The two Hay-Adams grand suites cost $1,600 a night, but Narva says the hotel will give a diplomatic discount to heads of state and their entourages. "Most of the other hotels extend a similar courtesy rate," she added. Former French president Giscard d'Estaing stayed at the Hay-Adams recently when here on a private visit.

The Regent Hotel, just opened, has two presidential suites, fireplaces, one to three bedrooms, as many as three baths, complete with a baby grand piano, all for $825 to $1,200. Queen Noor of Jordan, Saudi Prince Faisal Binsultan and Israeli Prime Minister Shimon Peres and his predecessor Yitzhak Shamir (now defense minister) have already tried out the suites, though not while on state visits.

The Embassy Row Hotel, also adding a suite, has had two rulers since it reopened after remodeling in September. Among other hotels that have recently rolled out the royal red carpet in the last two years or so, according to Roosevelt's office are the Watergate (six), Sheraton Washington (three), Washington Hilton (three), Hay-Adams (one) and Ritz-Carlton (one). Choosing the Hotel

"We don't tell them what hotel to go to," Roosevelt said. "We just give them a list of Washington hotels."

The decision as to where to stay is complicated. The country's ambassador to the United States and the visitor's advance team both customarily visit the accommodations in advance. In some cases, the visiting dignitary's staff has help. The Vista, for instance, this week is giving a party for 500 diplomats. Coyne, who has a huge refectory table full of pictures of himself with kings, presidents and many hardly lesser mortals, keeps in touch by letter and visit and runs the Madison as if it were his private guest house.

Bill Codus, a former assistant chief of protocol who now calls himself a "facilitator," used to take ambassadors to lunch at the Madison, diplomatically touting the hotel's desirability, while showing the suites, the reception rooms and the security. Now Codus works on a commission for the Vista. Like a travel agent, he takes a percentage of the bill, including the cost of room, food and drink. Guarding the Visitor

More important than the crystal chandeliers or the chef's trouble with truffles is the hotel's security. "Both the Madison and the Vista are side by side, with easy access to the White House and Congress," Codus says.

For a state visit the State Department and the Secret Service provide a command post of security people, along with representatives from the State Department protocol office, who offer hospitality.

Security begins with the best route to the hotel, according to William R. Gearhart, the security chief of the Four Seasons and president of the Hotel Security Directors Association.

When the state party arrives, Gearhart checks the lobby for suspicious characters. Then he goes through and makes sure that he can see everybody's hands. "We tell even the clerks behind the counter to be sure to put their hands on the top of the counter," Gearhart says. "And we make sure that the chief of state's party are the only ones in the elevator. When they have a conference, and they're really concerned, we have a bulletproof screen we put between them and the window." Suite Giving

Vista spends $10,000 a month for gifts for its special guests: free wine, champagne, chocolate, flowers, fruits, Givenchy toiletries, sewing kits, bathrobes. Every morning Braune hand-writes 20 cards to welcome incoming guests. Other hotels provide similar perks.

Hotels don't always plan the gifts some VIP visitors receive. Both the Four Seasons and the Madison admit to having guests who have considered the oriental rugs in the room in the same category as souvenir bath towels. Both hotels add that the culprits were not heads of state. Feeding the Famous

All the hotels say that the easiest part of the state visit is feeding the dignitary.

"Only one African leader -- I won't tell which -- brought all his own diet. Some rather unusual things. His chef tasted everything the king was to eat," Braune said. "Most like American food. It's highly publicized abroad. Their people all want milkshakes, cheeseburgers, Coca-Cola, even southern fried chicken."

Braune makes it his business to find out what the heads of state like. The French president drinks only red wine. Braune called a hotel friend in Mexico to learn that the Mexican president likes Bohemian beer.

The Four Seasons' concierge, Jack Nargil, once had to hold open a health-food store to get brownies for Queen Noor. The president of Portugal's young son celebrated his 10th birthday by eating his first pizza at the Vista. Surprises

Earlier this year, Prince Philip checked into the Embassy Row hotel with his family photographs (which he keeps on his bedside table) and asked for a dining room added to his suite, Peter Buse, the general manager, remembers.

When Italian President Bettino Craxi came, the Embassy Hotel installed a hot line from Rome to his suite. Every night the hot line rang. Every night, no one was on the other end. Finally the Italians got so mad, they yanked it out of the wall. Later they found out that in Rome every night, the cleaning woman picked up the phone to dust it.

The days (1957) when King Saud of Saudi Arabia gave gold watches to all the hotel chiefs and a car to the deputy chief of protocol are long gone. But Jordanian King Hussein and his wife, Queen Noor, did give watches to Nargil and Pat Gray, the presidential suite housekeeper.

What about the legendary dancing girls that potentates are said to wish provided?

"If they did ask for girls, we wouldn't tell you," said one hotel executive.