A360924 - OEIS

%I #35 Mar 02 2023 06:21:21

%S 0,2,3,4,4,5,5,6,6,6,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,

%T 11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,15,15,15,

%U 15,15,15,15,16,16,16,16,16,16,16,16,17,17,17,17,17,17,17,17,18,18,18

%N Smallest number of moves needed to win Integer Lunar Lander with starting position (0,n).

%C See A360923 for game rules.

%C Data provided by _Tom Karzes_.

%C It appears that a(n) = 1 + floor(sqrt(4*n-3)) for n>0 (which is essentially A000267 and A027434). - _N. J. A. Sloane_, Feb 25 2023 [This is proved by Casteigts, Raffinot, and Schoeters (2020) in the form a(n) = ceiling(2*sqrt(n)). - _Pontus von Brömssen_, Mar 01 2023]

%H Tom Karzes, <a href="/A360924/b360924.txt">Table of n, a(n) for n = 0..484</a>

%H Arnaud Casteigts, Mathieu Raffinot, and Jason Schoeters, <a href="https://arxiv.org/abs/2006.03666">VectorTSP: A Traveling Salesperson Problem with Racetrack-like acceleration constraints</a>, arXiv:2006.03666 [cs.DS], 2020. See Lemma 7.

%e From (0,6), a 5-move solution is (-1,5), (-2,3), (-2,1), (-1,0), (0,0). There is no shorter solution, so a(6) = 5.

%Y Top row of table A360923. Cf. A360925, A360926.

%Y See also A000267 and A027434.

%K nonn

%O 0,2

%A _Allan C. Wechsler_, Feb 25 2023