A065775 -id:A065775 - cp's OEIS frontend

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

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

A183043 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,0), (n,1),...,(n,n) on infinite chessboard.

Original entry on oeis.org

0, 3, 2, 2, 1, 4, 3, 2, 3, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 9, 8

Offset: 0

Clark Kimberling, Dec 20 2010

Stated another way, T(n,k) = distance from square (0,0) at center of an infinite open chessboard to square (n,k) via shortest knight path, for 0<=k<=n. - Fred Lunnon, May 18 2014

			Triangle starts:
0,
3,2,
2,1,4,
3,2,3,2,
2,3,2,3,4,
3,4,3,4,3,4,
4,3,4,3,4,5,4,
5,4,5,4,5,4,5,6,
4,5,4,5,4,5,6,5,6,
5,6,5,6,5,6,5,6,7,6
...
See examples under A242511.

Fred Lunnon, Knights in Daze, to appear.

Alois P. Heinz, Rows n = 0..140, flattened
Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A065775, A183044, A183045, A183046, A018837, A018839, A242511, A242512, A242513, A242514, A242591.

// See link for recursive & explicit algorithms. - Fred Lunnon, May 18 2014

See A065775.

Edited by N. J. A. Sloane, May 23 2014

Offset corrected by Alois P. Heinz, Sep 10 2014

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

A183053 Sums of knight's moves over the square |i|+|j|<=n on infinite chessboard.

Original entry on oeis.org

0, 12, 28, 48, 88, 148, 220, 312, 440, 588, 772, 1000, 1248, 1548, 1908, 2288, 2728, 3244, 3788, 4400, 5096, 5828, 6644, 7552, 8496, 9540, 10692, 11880, 13176, 14596, 16060, 17640, 19352, 21116, 23012, 25048

Offset: 0

Clark Kimberling, Dec 22 2010

nonn

Partial sums matching the squares |i|+|j|=n are given by A183052.

Cf. A065775, A183051, A183052.

See A065775.

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)^4*(x^2+1)*(x^2+x+1)^2). - Colin Barker, May 04 2014

A183050 Sums of knight's moves to points as in A183049.

Original entry on oeis.org

0, 3, 4, 5, 10, 15, 18, 23, 32, 37, 46, 57, 62, 75, 90, 95, 110, 129, 136, 153, 174, 183, 204, 227, 236, 261, 288, 297, 324, 355, 366, 395, 428, 441, 474, 509, 522, 559, 598, 611, 650, 693, 708, 749, 794, 811, 856, 903, 920, 969, 1020

Offset: 0

Clark Kimberling, Dec 22 2010

nonn

			a(3)=5=3+1+1, these summands being the least numbers of knight's moves from (0,0) to the points (3,0), (2,1), (1,2) on the 3rd diagonal in the 1st quadrant - which is 1/4 of a 3rd concentric square about the origin.  See A183052 for sums over the concentric squares.

Cf. A065775, A183049, A183051.

See A065775.

Empirical g.f.: 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

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.

A183045 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,-n), (n,-n+1), ... (n,n) on infinite chessboard.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 20 2010

			First five rows:
0
2 3 2
4 1 2 1 4
2 3 2 3 2 3 2
4 3 2 3 2 3 2 3 4

Cf. A065775, A183043.

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

A183042 Least number of knight's moves from (0,0) to the segment of points (0,n), (1,n-1), ..., (n,0) on infinite chessboard.

Original entry on oeis.org

0, 6, 6, 8, 12, 18, 22, 28, 36, 42, 52, 64, 68, 82, 98, 104, 118, 138, 146, 164, 184, 194, 216, 240, 248, 274, 302, 312, 338, 370, 382, 412, 444, 458, 492, 528, 540, 578, 618, 632, 670, 714, 730, 772, 816, 834, 880, 928, 944, 994

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

			For n=3, the least number of knight's moves to the points (i.e., squares) (3,0), (2,1), (1,2), (0,3) are 3,1,1,3, respectively, for a total of a(3)=8.

Cf. A065775.

a(n)=T(n,0)+T(n-1,1)+...+T(0,n), where T is formulated at A065775.

Empirical g.f.: 2*x*(x^13-x^9-3*x^7-x^6-4*x^2-3*x-3) / ((x-1)^3*(x+1)*(x^2+1)*(x^2+x+1)^2). - Colin Barker, May 04 2014

A183044 Sums of least numbers of knight's moves on vertical segments (n,0) to (n,n) on infinite chessboard.

Original entry on oeis.org

0, 5, 7, 10, 14, 21, 27, 38, 44, 57, 67, 82, 92, 111, 123, 144, 158, 181, 197, 224, 240, 269, 289, 320, 340, 375, 397, 434, 458, 497, 523, 566, 592, 637, 667, 714, 744, 795, 827, 880, 914, 969, 1005, 1064, 1100, 1161, 1201, 1264

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

Row sums of A183043.

Cf. A065775, A183043.

Empirical g.f.: x*(2*x^10+2*x^9-2*x^8-4*x^7-5*x^6-x^5+x^4-2*x^3-5*x^2-7*x-5) / ((x-1)^3*(x+1)^2*(x^2+1)*(x^2+x+1)). - Colin Barker, May 04 2014

cp's OEIS Frontend

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

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A183043 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,0), (n,1),...,(n,n) on infinite chessboard.

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

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A183053 Sums of knight's moves over the square |i|+|j|<=n on infinite chessboard.

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A183050 Sums of knight's moves to points as in A183049.

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

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A183045 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,-n), (n,-n+1), ... (n,n) on infinite chessboard.

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

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A183042 Least number of knight's moves from (0,0) to the segment of points (0,n), (1,n-1), ..., (n,0) on infinite chessboard.

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