A183053 -id:A183053 - cp's OEIS frontend

A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

0, 3, 3, 2, 2, 2, 3, 1, 1, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 5, 3, 3, 3, 3, 3, 3, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 6, 6, 4, 4, 4, 4, 4, 4, 4, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 6, 6, 6, 6, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7

Offset: 0

Stewart Gordon, Dec 05 2001

			From _Clark Kimberling_, Dec 20 2010: (Start)
T(i,j) for -2<=i<=2 and -2<=j<=2:
  4 1 2 1 4=T(2,2)
  1 2 3 2 1=T(2,1)
  2 3 0 3 2=T(2,0)
  1 2 3 2 1=T(2,-1)
  4 1 2 1 4=T(2,-2)
Corner of the array, T(i,j) for i>=0, j>=0: [Corrected Oct 14 2016]
  0 3 2 3 2 3 4...
  3 2 1 2 3 4 3...
  2 1 4 3 2 3 4...
  3 2 3 2 3 4 2... (End)

Identical to A049604 except for T(1, 1).

For number of knight's moves to various subsets of the chessboard, see A018837, A183041-A183053.

From Clark Kimberling, Dec 20 2010: (Start)
T(i,j) is given in cases:
Case 1:  row 0
  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.
Case 2:  row 1
  T(0,1)=3, T(1,1)=2, and for m>=1,
  T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1,
  T(4m+1,1)=2m+2.
Case 3:  columns 0 and 1
  (column 0 = row 0); (column 1 = row 1).
Case 4:  For i>=2 and j>=2,
   T(i,j)=1+min{T(i-2,j-1),T(i-1,j-2)}.
Cases 1-4 determine T in the 1st quadrant;
all other T(i,j) are easily obtained by symmetry. (End)

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

Original entry on oeis.org

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

N. J. A. Sloane, Marc LeBrun

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

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

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).

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

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

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 *)

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

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

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

A183041 Least number of knight's moves from (0,0) to (n,1) on infinite chessboard.

Original entry on oeis.org

3, 2, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 9, 10, 9, 10, 11, 12, 11, 12, 13, 14, 13, 14, 15, 16, 15, 16, 17, 18, 17, 18, 19, 20, 19, 20, 21, 22, 21, 22, 23, 24, 23, 24, 25, 26, 25, 26, 27, 28, 27, 28, 29, 30, 29, 30, 31, 32, 31, 32, 33, 34, 33, 34, 35, 36

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

Row 2 of the array at A065775.

Apparently a(n)=A162330(n), n>0. - R. J. Mathar, Jan 29 2011

			a(0)=3 counts (0,0) to (2,1) to (1,3) to (0,1).

Cf. A065775, A018837, A183042-A183053.

def a(n):
  if n < 2: return [3, 2][n]
  m, r = divmod(n, 4)
  return [2*m+1, 2*m+2][r%2]
print([a(n) for n in range(70)]) # Michael S. Branicky, Mar 02 2021

T(0,1)=3, T(1,1)=2, and for m>=1,
T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1, T(4m+1,1)=2m+2.
G.f.: (2*x^5-2*x^4+x^3-x^2-x+3) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 19 2014

A183051 Array of least knight's moves to points on the square |i|+|j|=n on infinite chessboard.

Original entry on oeis.org

0, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 2, 4, 5, 3, 3, 3, 3, 3, 3

Offset: 1

Clark Kimberling, Dec 22 2010

			Top 5 rows:
0
3 3 3 3
2 2 2 2 2 2 2 2
3 1 1 3 1 1 3 1 1 3 1 1
2 2 4 2 2 2 4 2 2 2 4 2 2 2 4 2
Row n has 4n numbers which form a square of points (i.e., unit squares) on an infinite chessboard.  The first 3 of these concentric squares are represented as follows:
....2
..2.3.2
2.3.0.3.2
..2.3.2
....2

Cf. A065775, A183050, A183053.

See A065775.

A183052 Sums of knight's moves from (0,0) to points on the square |i|+|j|=n on infinite chessboard.

Original entry on oeis.org

0, 12, 16, 20, 40, 60, 72, 92, 128, 148, 184, 228, 248, 300, 360, 380, 440, 516, 544, 612, 696, 732, 816, 908, 944, 1044, 1152, 1188, 1296, 1420, 1464, 1580, 1712, 1764, 1896, 2036, 2088, 2236, 2392, 2444, 2600, 2772

Offset: 0

Clark Kimberling, Dec 22 2010

nonn

Partial sums of A183053, which counts knight's moves from (0,0) to all points (i,j) such that |i|+|j|<=n.

			0=0
12=3+3+3+3
16=2+2+2+2+2+2+2+2
20=3+1+1+3+1+1+3+1+1+3+1+1
40=4*(3+3+3+3+3)

Cf. A065775, A183050, A183051, A183053.

See A065775.
a(n) = 4*A183050(n).
Empirical g.f.: 4*x*(2*x^12-2*x^11+2*x^10-4*x^9+2*x^8-x^7-x^6-4*x^4-4*x^2-x-3) / ((x-1)^3*(x^2+1)*(x^2+x+1)^2). - Colin Barker, May 04 2014

cp's OEIS Frontend

A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

Views

Author

Keywords

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A183041 Least number of knight's moves from (0,0) to (n,1) on infinite chessboard.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A183051 Array of least knight's moves to points on the square |i|+|j|=n on infinite chessboard.

Views

Author

Keywords

Examples

Crossrefs

Formula

A183052 Sums of knight's moves from (0,0) to points on the square |i|+|j|=n on infinite chessboard.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula