A018837 Number of steps for knight to reach (n,0) on infinite chessboard.

0, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38, 39, 38, 39, 40, 41, 40, 41, 42, 43

Offset: 0

Author

N. J. A. Sloane, Marc LeBrun

Keywords

• nonn
• easy

Comments

The knight starts at (0,0) and we count the least number of steps. Row 1 of the array at A065775. - Clark Kimberling, Dec 20 2010

Apparently also the minimum number of steps of the (1,3)-leaper to reach (n,n) starting at (0,0). - R. J. Mathar, Jan 05 2018

Examples

			a(1)=3 counts these moves: (0,0) to (2,1) to (0,2) to (1,0). - _Clark Kimberling_, Dec 20 2010
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Francis N. Castro, Oscar E. González and Luis A. Medina, Generalized exponential sums and the power of computers, Involve, Vol. 11 (2018), Issue 1, pp. 127-142. Also, authors' copy.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A065775, A183041-A183053, A083219 (essentially the same).

Cf. A018840 for the (2,3)-leaper.

Programs

Mathematica

CoefficientList[Series[x (3 - x + x^2 - x^3 - 2 x^4 + 2 x^5)/((1-x)^2 (1+x) (1+x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2018 *)
Array[Which[#==1,3,True,(#+Mod[#,4])/2]&,100,0] (* Elisha Hollander, Aug 05 2021 *)

PARI

concat([0], Vec( x*(3-x+x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1+x)*(1+x^2)) + O(x^166) ) ) \\ Joerg Arndt, Sep 10 2014

Python

def a(n): return 3 if n == 1 else (n + n % 4) // 2 # Elisha Hollander, Aug 05 2021

Formula

a(n) = 2[ (n+2)/4 ] if n even, 2[ (n+1)/4 ]+1 if n odd (n >= 8).

G.f.: x*(3-x+x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1+x)*(1+x^2)). a(n)=A083219(n), n<>1. - R. J. Mathar, Dec 15 2008

T(0,0)=0, T(1,0)=3, and for m>=1, T(4m-2,0)=2m, T(4m-1,0)=2m+1, T(4m,0)=2m, T(4m+1,0)=2m+1 where T(.,.) = A065775(.,.). - Clark Kimberling, Dec 20 2010

Sum_{n>=1} (-1)^n/a(n) = 5/3 - 2*log(2). - Amiram Eldar, Sep 10 2023