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.

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

Author

Fred Lunnon, May 16 2014 and May 18 2014

Keywords

• easy
• nonn
• walk

Comments

[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.

Examples

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

References

• Fred Lunnon, Knights in Daze, to appear.

Links

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

Crossrefs

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

Programs

Magma

[ 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] ];

PARI

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

Formula

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 ).