NEW ORLEANS — Like a train on a transcontinental route, Anthony Davis has been on his way all season.

His 18 points, 15 rebounds and two blocks in Friday’s victory against Memphis served as another impressive stop on his rookie-season trip which, as it nears its end, seems to have gotten better and better with time.

“It’s been great so far,” said Davis, the No. 1 pick in the 2012 draft. “A couple of injuries. Having fun. Trying to explore how to establish myself in this league.”

That he has improved was to be expected. But it’s more about how he has gotten better — or the myriad ways he has shown he can contribute — that has been impressive and makes for an intriguingly optimistic future.

Early on, Davis, who’s averaging 13.2 points and 8.0 rebounds, flashed the kind of defense and rebounding — with some scoring — that the Hornets thought he would bring and which makes him a great fit for coach Monty Williams. There were glimpses of more, such as his 28 points and 11 rebounds Nov. 17 at Milwaukee.

“I think he’s starting to understand the different game plans and disciplines,” Hornets top assistant Randy Ayers said. “One night, he could be guarding (Boston’s Kevin) Garnett, the next night (Portland’s) LaMarcus Aldridge and a third night someone with a completely different skill set. He’s adjusting to the schedule, going against different guys and how quickly that changes.”

Near the midpoint of the season, his ability to be effective running the court — scoring and putting pressure on the opponent’s transition defense — became more evident. Then, during the All-Star Weekend Rising Stars Challenge, he brandished a jump shot like a shiny new toy.

He had shown a glimpse of that as far back as the Olympics, when he made a 3-pointer, then turned and looked at the opposing team’s coach. That happened to be his college coach, John Calipari of Kentucky moonlighting for the Dominican Republic, who didn’t want him shooting it.

“It’s something I’ve always been able to do,” Davis said, “but I didn’t want to take it unless it was part of our offensive game plan.”

Two weeks ago, it all seemed to start coming together for Davis, who began posting games of double-figure scoring and rebounding more frequently. From March 4-12, he had double-doubles in four of five games and was well on his way to one in the next game at Washington.

“I was just going out, playing hard with a lot of energy, not just settling for being out there — playing hard like Coach Mont wants me to,” he said. “When I do that, it’s a lot easier to get double-doubles.”

Williams, demanding of Davis because of his tremendous potential, has been the 20-year-old’s harshest critic at times, but he also often points out his young star’s inexperience. Asked about Davis’ recent collection of double-doubles, Williams said he should be doing that more consistently. He pointed to the game March 15 at Washington, a 96-87 loss. Hampered by five fouls, Davis had 16 points and seven rebounds in just 16 minutes.

“He was on his way to a monster game but couldn’t stay out of foul trouble,” said Williams, obviously feeling that, had Davis’ performance continued, the Hornets would have won.

One game after his tip-in with 0.3 seconds left beat Boston on Wednesday, Davis had his big game vs. Memphis. That came immediately after Williams said the rail-thin rookie “is going to have to live in the weight room this summer” and mentioned how Chicago’s Carlos Boozer smashed him into the basket support in a game Feb. 19 at New Orleans Arena.

It has been evident since his lone year at Kentucky that Davis, at 6-foot-11 and 220 pounds, would have to get stronger to compete against NBA power forwards. This season, the more muscular ones sometimes have been a difficult matchup.

Last Monday at the Arena, Golden State All-Star David Lee had success backing down Davis in the post and scoring. And Davis has had problems with the bulk and strength of players who aren’t All-Stars — such as Utah reserve Derrick Favors, who’s 6-10, 258 and athletic.

But if Davis lacks physical strength, he has shown he has another kind.

As if to prove a point Friday against Memphis, he put his body on the line against power forward Zach Randolph and outplayed him, holding him to 14 points and nine rebounds while taking a shot to the face that comes with the territory of standing your ground. With 1:33 left and the Hornets clinging to an 86-81 lead, Davis forced Randolph to miss in the low post, then wrestled for and got the rebound.

Afterward, Hornets point guard Greivis Vasquez lauded Davis.

“This kid, Anthony Davis, is going to be a player,” he said. “He did that against Zach Randolph? I know Zach; I played with him for a year, and that guy is a monster. And he was frustrated with AD.”

Said Randolph: “His length causes problems (defensively), and now he’s shooting the ball, has a little face-up jumper, trying to drive, and he’s doing all that at 7 feet. His upside is scary. He’ll be an All-Star real soon.”

It wasn’t the first time Davis had gotten the better of one of the league’s best. Ahead of Portland coming to town Feb. 13, Williams gushed about Aldridge, whom he had tutored as an assistant with the Trail Blazers, going as far to say he thinks he’s the best power forward in the league.

In that game, though, Davis put on a show on the fast break, caught lobs in the set offense and dunked putbacks on his way 21 points and 11 rebounds in a 99-63 win. He helped hold Aldridge to six points and three rebounds.

Just 33 games into the season, Williams suggested Davis may have hit the rookie wall after four games of single-digit scoring. But on Jan. 7 against San Antonio, the best team in the Southwest Division, Davis had 17 points, nine rebounds (four offensive), three steals and a blocked shot in a 95-88 victory.

Some of his best efforts have come after injuries. After missing two games with a mild concussion, he had 23 points and 11 rebounds in a win against Charlotte. Returning after missing 11 games with a stress reaction in his left ankle, he reeled off 10 games in a row of double-digit scoring, including three with double-doubles and two with nine rebounds. The stretch of double-doubles earlier this month came after he missed two games with a shoulder sprain.

Portland point guard Damian Lillard is certain to be selected Rookie of the Year, but Grizzlies coach Lionel Hollins said Davis would be in strong contention if not for injuries.

“He makes some shots, he runs the floor, he attacks the offensive glass, he blocks shots and he rebounds,” Hollins said.

One weakness in Davis’ game, due in part to his lack of size and strength, is that he is not yet a good post-up player. Asked whether that was a challenge for Davis, Hollins said, “I don’t think it’s a challenge at all. I think it’s a maturation. He’s going to be an offensive player in time.”

With the season nearing its end, Williams said Davis, after taking two weeks off, has a summer of vigorous training ahead — mostly lifting weights and doing a lot of shooting. Teammate Jason Smith, who said his body looked just like Davis’ when he was that age, said Davis will gain muscle naturally during the next three years.

“He’s already lifting now, and he’s real conscious about eating right and drinking protein shakes, so the weight will come on right,” Smith said.

Davis already has been more than a ready and willing worker, which is what Ayers said has been key to his development.

“The biggest thing I got better at this season was just getting with Coach Mont and the assistant coaches and learning the game,” Davis said. “Lately, I’ve been working on just trying to finish this season strong. I look forward to working hard this summer to make myself better.”