A018838 Minimum number of steps for a knight to reach (n,n) on an infinite chessboard.

0, 2, 4, 2, 4, 4, 4, 6, 6, 6, 8, 8, 8, 10, 10, 10, 12, 12, 12, 14, 14, 14, 16, 16, 16, 18, 18, 18, 20, 20, 20, 22, 22, 22, 24, 24, 24, 26, 26, 26, 28, 28, 28, 30, 30, 30, 32, 32, 32, 34, 34, 34, 36, 36, 36, 38, 38, 38, 40, 40, 40, 42, 42, 42, 44, 44, 44

Offset: 0

Author

N. J. A. Sloane, Marc LeBrun

Keywords

• nonn
• easy
• nice

Comments

Apparently also the minimum number of steps of the (1,3)-leaper to reach (2n,0) starting at (0,0). The (1,3)-leaper cannot reach (2n+1,0). - R. J. Mathar, Jan 05 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A002264.

Programs

Magma

[0,2,4] cat [2*Ceiling(n/3): n in [3..80]]; // Vincenzo Librandi, Oct 16 2013

Mathematica

Join[{0,2,4},Table[2*Ceiling[n/3],{n,3,70}]] (* Harvey P. Dale, Jul 27 2012 *)
CoefficientList[Series[2 x (x^5 - x^4 - x^2 + x + 1)/((x - 1)^2 (x^2 + x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 16 2013 *)
LinearRecurrence[{1,0,1,-1},{0,2,4,2,4,4,4},70] (* Harvey P. Dale, Nov 03 2019 *)

PARI

a(n)=if(n>2, (n+2)\3*2, 2*n) \\ Charles R Greathouse IV, Feb 10 2017

Formula

a(n) = 2*ceiling(n/3) = 2*A002264(n+2), n >= 3.

G.f.: 2*x*(x^5-x^4-x^2+x+1)/((x-1)^2*(x^2+x+1)). - Colin Barker, Oct 04 2012

a(n) = a(n-1) + a(n-3) - a(n-4). - Wesley Ivan Hurt, Feb 21 2025