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

Author

Fred Lunnon, May 16 2014 and May 18 2014

Keywords

• easy
• nonn
• walk

Comments

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.

Examples

			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).
		

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. A242512, A242513, A242514, A183043, A242591.

Programs

Magma

[ 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] ];

Maple

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

Mathematica

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 *)

Formula

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