Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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