A360923 -id:A360923 - cp's OEIS frontend

A361070 a(n) is the number of occurrences of n in A360923.

1, 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, 21, 27, 30, 35, 40, 44, 52, 56, 63, 70, 75, 85, 90, 100, 107, 115, 126, 132, 145, 153, 163, 175, 182, 199, 206, 220, 232, 242, 259, 268, 285, 297, 310, 328, 337, 359, 370, 387, 404, 416, 440, 451, 472, 489, 504, 528, 540

Offset: 0

Rémy Sigrist, Mar 01 2023

nonn

For n > 0, the number of starting positions from Z^2 at distance n from (0, 0) appears to be A051890(n):
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
  . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 6 6 6 6
  . . . . . . . . . . . . . . . . . . . . 6 4 5 5 5 5 6 6 6 6 6
  . . . . . . . . . . . . . . . . 6 5 3 4 4 4 5 5 5 5 6 6 6 6 .
  . . . . . . . . . . . . 6 6 5 4 2 3 3 4 4 4 5 5 5 6 6 6 6 . .
  . . . . . . . . . 6 6 5 5 4 3 1 2 3 3 4 4 5 5 5 6 6 6 . . . .
  . . . . . . 6 6 6 5 5 4 4 3 2 0 2 3 4 4 5 5 6 6 6 . . . . . .
  . . . . 6 6 6 5 5 5 4 4 3 3 2 1 3 4 5 5 6 6 . . . . . . . . .
  . . 6 6 6 6 5 5 5 4 4 4 3 3 2 4 5 6 6 . . . . . . . . . . . .
  . 6 6 6 6 5 5 5 5 4 4 4 3 5 6 . . . . . . . . . . . . . . . .
  6 6 6 6 6 5 5 5 5 4 6 . . . . . . . . . . . . . . . . . . . .
  6 6 6 6 6 5 . . . . . . . . . . . . . . . . . . . . . . . . .
  6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The largest column of A360923 containing n appears to have index A002620(n).

			Square array A360923 begins as follows:
     0 2 3 4 4 5 5 6 6 6 7 7 7 8 8 8 8 .
     1 3 4 5 5 6 6 7 7 7 8 8 8 . . . . .
     4 5 6 6 7 7 8 8 8 . . . . .
     7 7 8 8 . . . . . .
     . . . . .
Hence a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 5, a(6) = 7, a(7) = 10 and a(8) = 12.

Rémy Sigrist, C++ program

Cf. A002620, A051890, A360923.

A360924 Smallest number of moves needed to win Integer Lunar Lander with starting position (0,n).

Original entry on oeis.org

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, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18

Offset: 0

Allan C. Wechsler, Feb 25 2023

nonn

See A360923 for game rules.
Data provided by Tom Karzes.
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]

			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.

Tom Karzes, Table of n, a(n) for n = 0..484
Arnaud Casteigts, Mathieu Raffinot, and Jason Schoeters, VectorTSP: A Traveling Salesperson Problem with Racetrack-like acceleration constraints, arXiv:2006.03666 [cs.DS], 2020. See Lemma 7.

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

A360925 Smallest number of moves needed to win Integer Lunar Lander from starting position (n,0).

Original entry on oeis.org

0, 1, 4, 7, 9, 12, 14, 17, 19, 21, 24, 26, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 55, 58, 60, 63, 65, 67, 70, 72, 75, 77, 79, 82, 84, 87, 89, 92, 94, 96, 99, 101, 104, 106, 108, 111, 113, 116, 118, 120, 123, 125, 128, 130, 133, 135, 137, 140, 142, 145

Offset: 0

Allan C. Wechsler, Feb 25 2023

nonn

See A360923 for game rules.
Data from Tom Karzes.
The first differences begin 1, 3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, ... Are all the differences after the start either 2 or 3? - N. J. A. Sloane, Feb 25 2023
Conjecture: For n >= 2, a(n) =  n+1+floor(sqrt(2*n^2-2*n-3)). - N. J. A. Sloane, Feb 26 2023

			For starting position (3,0), a 7-move solution is (2,2), (1,3), (0,3), (-1,2), (-1,1), (-1,0), (0,0). There are no shorter solutions, so a(3) = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..250
Rémy Sigrist, C++ program

First column of table A360923.

Cf. A360924, A360926.

More terms from Rémy Sigrist, Feb 26 2023

A360926 Smallest number of moves needed to win Integer Lunar Lander with a starting position of (n,n).

Original entry on oeis.org

0, 3, 6, 8, 11, 13, 16, 18, 20, 23, 25, 28, 30, 33, 35, 37, 40, 42, 45, 47, 49, 52, 54, 57, 59, 62, 64, 66, 69, 71, 74, 76, 78, 81, 83, 86, 88, 91, 93, 95, 98, 100, 103, 105, 107, 110, 112, 115, 117, 119, 122, 124, 127, 129, 132, 134, 136, 139, 141, 144, 146

Offset: 0

Allan C. Wechsler, Feb 25 2023

nonn

See A360923 for the rules of Integer Lunar Lander.
Data from Tom Karzes.
The conjectured formula for A360923 implies a formula for a(n). - N. J. A. Sloane, Feb 26 2023

			From starting position (3,3), an 8-move solution is (2,5), (1,6), (0,6), (-1,5), (-2,3), (-2,1), (-1,0), (0,0). There is no shorter solution, so a(3) = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..300
Rémy Sigrist, C++ program

Main diagonal of table A360923. Cf. A360924, A360925.

More terms from Rémy Sigrist, Feb 26 2023

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A361070 a(n) is the number of occurrences of n in A360923.

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A360924 Smallest number of moves needed to win Integer Lunar Lander with starting position (0,n).

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A360925 Smallest number of moves needed to win Integer Lunar Lander from starting position (n,0).

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A360926 Smallest number of moves needed to win Integer Lunar Lander with a starting position of (n,n).

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