A183043 -id:A183043 - cp's OEIS frontend

A242511 a(n) = number of knight's move paths of minimal length n steps, from origin (0,0) at center of an infinite open chessboard to square (0,0) for n=0; square (2,-1) for n=1; and square (2n-3, (n+1)mod 2) for n>=2.

1, 1, 2, 6, 28, 100, 330, 1050, 3024, 8736, 23220, 62700, 158004, 406692, 986986, 2452450, 5788640, 14002560, 32357052, 76640148, 174174520, 405623400, 909582212, 2089064516, 4633556448, 10519464000, 23120533800, 51977741400, 113365499940, 252725219460, 547593359850, 1211884139250, 2610998927040, 5741708459520, 12309472580460, 26917328938500, 57457069777800, 125016198060600, 265832233972140, 575824335603660, 1220234181784800

Offset: 0

Fred Lunnon, May 16 2014 and May 18 2014

The squares concerned constitute an infinite, locally fully concertinaed knight's path from the origin, which hugs the axis y=0 and is minimal to each square.

			For n=0 there is a(0)=1 path from (0,0) to (0,0) with 0 step.
For n=1 there is a(1)=1 path from (0,0) to (2,-1) with 1 step.
For n=2 there are a(2)=2 paths from (0,0) to (1,1) with 2 steps:
  (0,0) -> (2,-1) -> (1,1) and (0,0) -> (-1,2) -> (1,1).
For n=3 there are a(3)=6 paths from (0,0) to (3,0) with 3 steps:
  (0,0)(2,-1)(1,1)(3,0); (0,0)(2,1)(1,-1)(3,0); (0,0)(2,-1)(4,-2)(3,0);
  (0,0)(2,1)(4,2)(3,0); (0,0)(-1,-2)(1,-1)(3,0); (0,0)(-1,2)(1,1)(3,0).

Fred Lunnon, Knights in Daze, to appear.

Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A242512, A242513, A242514, A183043, A242591.

[ Max(1, Binomial(d, d div 2 - 1)/6 * // axis-hugging path
  ( /*if*/ IsEven(d) select (d^2-2*d+6)*(d^2+8)/(d+4)
  else (d-1)*(d^2-2*d+15) /*end if*/ )) : d in [0..20] ];

A242511 := proc(n)
    local a;
    if n <=1 then
        return 1;
    end if ;
    a := binomial(n,floor(n/2)-1)/6 ;
    if type(n,'even') then
        a*(n^2-2*n+6)*(8+n^2)/(n+4) ;
    else
        a*(n-1)*(n^2-2*n+15) ;
    end if ;
end proc: # R. J. Mathar, May 17 2014

q := (1 - 2 x)^(7/2) (1 + 2 x)^(5/2); CoefficientList[Series[(-10 + 10 x + 127 x^2 - 111 x^3 - 576 x^4 + 410 x^5 + 1072 x^6 - 528 x^7 - 624 x^8 + 144 x^9 + q (10 + 10 x - 7 x^2 - 3 x^3 + x^4 + x^5))/(q*x^4), {x, 0, 20}],x] (* Benedict W. J. Irwin, Oct 20 2016 *)

For n>=2, a(n) = binomial(n,floor(n/2)-1)/6 *
      ( (n^2-2*n+6)*(n^2+8)/(n+4) if n even, (n-1)*(n^2-2*n+15) if n odd ).
G.f.: (-10 + 10*x + 127*x^2 - 111*x^3 - 576*x^4 + 410*x^5 + 1072*x^6 - 528*x^7 - 624*x^8 + 144*x^9 + q*(10 + 10*x - 7*x^2 - 3*x^3 + x^4 + x^5))/(q*x^4), where q = sqrt((1 - 2*x)^7*(1 + 2*x)^5). - Benedict W. J. Irwin, Oct 20 2016

A242514 a(n) is the maximal number of shortest knight's move paths, from origin at center of an infinite open chessboard, to square with coordinates <= n.

Original entry on oeis.org

1, 12, 54, 54, 54, 54, 85, 240, 240, 588, 1512, 1512, 3564, 8700, 8700, 19965, 47124, 47124, 105963, 244244, 244244, 540540, 1224080, 1224080, 2674984, 5974956, 5974956, 12924522, 28553200, 28553200, 61250490, 134104432, 134104432, 285689624, 620826672, 620826672, 1314933000, 2839363800, 2839363800, 5984393805, 12852021420, 12852021420, 26973910215, 57655813500, 57655813500, 120569654700, 256649540640, 256649540640, 535009931280, 1134692142540, 1134692142540, 2358818719950, 4986548028000, 4986548028000, 10340761857030, 21796919253120, 21796919253120, 45102668144040, 94821703158000, 94821703158000, 195825873726600, 410720543218440, 410720543218440, 846739738410930, 1772108740270440, 1772108740270440, 3647615648094990, 7618942347630120, 7618942347630120, 15660031688889048, 32650847564232672

Offset: 0

Fred Lunnon, May 16 2014 and May 18 2014

For n > 5 the distinct terms of this sequence are conjectured to be identical to A242512: precisely, A242514(n) = A242512(ceiling(2*(n+1)/3)).

			For n=7, there are 240 shortest paths of length 6 steps from (0,0) to (7,7);
no square within 0 <= x,y <= 7 has more shortest paths.

Fred Lunnon, Knights in Daze, to appear.

Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A242511, A242512, A242513, A183043, A242591.

A242591 Triangle of number of shortest knight paths T(n,k) from square (0,0) at center of an infinite open chessboard to square (n,k), for 0 <= k <= n.

Original entry on oeis.org

1, 12, 2, 2, 1, 54, 6, 2, 9, 2, 2, 6, 1, 3, 32, 6, 28, 6, 24, 3, 8, 24, 3, 18, 1, 12, 85, 6, 100, 16, 95, 12, 60, 4, 25, 240, 6, 70, 4, 50, 1, 30, 201, 10, 60, 40, 330, 35, 266, 20, 150, 5, 66, 588, 20, 210, 10, 180, 5, 120, 1, 60, 462, 15, 147, 1512

Offset: 0

Fred Lunnon, May 18 2014

			Triangle starts:
    1;
   12,   2;
    2,   1, 54;
    6,   2,  9,   2;
    2,   6,  1,   3, 32;
    6,  28,  6,  24,  3,   8;
   24,   3, 18,   1, 12,  85,   6;
  100,  16, 95,  12, 60,   4,  25, 240;
    6,  70,  4,  50,  1,  30, 201,  10,  60;
   40, 330, 35, 266, 20, 150,   5,  66, 588, 20;
  ...
See examples under A242511.

Georg Fischer, Table of n, a(n) for n = 0..209
Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A242511, A242512, A242513, A242514, A183043.

// See attached a-file for recursive & explicit algorithms.

a(66) ff. exported to b-file by Georg Fischer, Jul 16 2020

A242512 a(n) = number of knight's move paths of minimal length n steps, from origin at center of an infinite open chessboard to square (0,0) for n=0; to square (2,-1) for n=1; and to square ([(3n-3)/2], [(3n-4)/2]) for n>=2.

Original entry on oeis.org

1, 1, 2, 9, 32, 85, 240, 588, 1512, 3564, 8700, 19965, 47124, 105963, 244244, 540540, 1224080, 2674984, 5974956, 12924522, 28553200, 61250490, 134104432, 285689624, 620826672, 1314933000, 2839363800, 5984393805, 12852021420, 26973910215, 57655813500, 120569654700, 256649540640, 535009931280, 1134692142540, 2358818719950, 4986548028000, 10340761857030, 21796919253120, 45102668144040, 94821703158000

Offset: 0

Fred Lunnon, May 16 2014 and May 18 2014

[x] denotes floor(x), the largest integer <= x. E.g., [-1/2] = -1.

The squares concerned constitute an infinite, locally fully concertinaed knight path from the origin, which hugs the diagonal x=y and is minimal to each square.

			See also examples for A242511.
For n=3, there are a(3)=9 minimal paths of 3 steps from (0,0) to (3,2).

Fred Lunnon, Knights in Daze, to appear.

Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A242511, A242513, A242514, A183043, A242591.

[ Max(1, Binomial(d, d div 2 - 1)/2 * // diagonal-hugging path
  ( /*if*/ IsEven(d) select (d^3-d^2+30*d-40)/(d+4)
  else d*(d^2+2*d+33)/(d+5) /*end if*/ )) : d in [0..20] ];

a(n) = max(1, binomial(n, (n\2 - 1))/2 * if (n%2, n*(n^2+2*n+33)/(n+5), (n^3-n^2+30*n-40)/(n+4))); \\ Michel Marcus, May 17 2014

For n>=2, a(n) = binomial(n,[n/2]-1)/2 *

( (n^3-n^2+30n-40)/(n+4) if n even, n(n^2+2n+33)/(n+5) if n odd ).

A242513 a(n) = maximal number of shortest knight's move paths, from origin at center of an infinite open chessboard, to any square within n moves.

Original entry on oeis.org

1, 1, 2, 12, 54, 100, 330, 1050, 3024, 8736, 23220, 62700, 158004, 406692, 986986, 2452450, 5788640, 14002560, 32357052, 76640148, 174174520, 405623400, 909582212, 2089064516, 4633556448, 10519464000, 23120533800, 51977741400, 113365499940, 252725219460, 547593359850, 1211884139250, 2610998927040, 5741708459520, 12309472580460, 26917328938500

Offset: 0

Fred Lunnon, May 16 2014 and May 18 2014

For n>4 this sequence is conjectured to be identical to A242511.

The same sequence results after replacing 'within n moves' with 'at shortest distance n moves'.

			For n=5, there are 100 shortest paths of length 5 steps from (0,0) to (7,0); no square at 5 (or fewer) moves from the origin has more shortest paths.

Fred Lunnon, Knights in Daze, to appear.

Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A242511, A242512, A242514, A183043, A242591.

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.

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

A242511 a(n) = number of knight's move paths of minimal length n steps, from origin (0,0) at center of an infinite open chessboard to square (0,0) for n=0; square (2,-1) for n=1; and square (2n-3, (n+1)mod 2) for n>=2.

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A242514 a(n) is the maximal number of shortest knight's move paths, from origin at center of an infinite open chessboard, to square with coordinates <= n.

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A242591 Triangle of number of shortest knight paths T(n,k) from square (0,0) at center of an infinite open chessboard to square (n,k), for 0 <= k <= n.

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A242512 a(n) = number of knight's move paths of minimal length n steps, from origin at center of an infinite open chessboard to square (0,0) for n=0; to square (2,-1) for n=1; and to square ([(3n-3)/2], [(3n-4)/2]) for n>=2.

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A242513 a(n) = maximal number of shortest knight's move paths, from origin at center of an infinite open chessboard, to any square within n moves.

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

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