A166540 Number of ways to place 2 nonattacking kings on an n X n X n raumschach board.

0, 0, 0, 193, 1548, 6714, 21280, 55395, 125748, 257908, 489024, 870885, 1473340, 2388078, 3732768, 5655559, 8339940, 12009960, 16935808, 23439753, 31902444, 42769570, 56558880, 73867563, 95379988, 121875804, 154238400, 193463725, 240669468, 297104598, 364159264

Offset

0,4

Comments

We consider that kings "attack" any square that differs by at most one in any combination of the indices from its current space. A logical extension of sequence A061995.

Links

Andrew Woods, Table of n, a(n) for n = 0..1000

Wikipedia, Three-dimensional chess

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (n^6 - (3*n-2)^3) / 2 = (n^6)/2 - (27*n^3)/2 + 27*n^2 - 18*n + 4 for n>0. - Andrew Woods, Aug 30 2011

G.f.: x^3*(193 +197*x -69*x^2 +35*x^3 +4*x^4)/(1-x)^7. - Colin Barker, Jan 09 2013

E.g.f.: (8 -8*x +4*x^2 +63*x^3 +65*x^4 +15*x^5 +x^6)*exp(x)/2 -4. - G. C. Greubel, Apr 03 2019

Mathematica

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 193, 1548, 6714, 21280, 55395}, 35] (* G. C. Greubel, May 16 2016; a(7) appended by Georg Fischer, Apr 03 2019 *)

Prog

(Python) # Two non-attacking kings in n x n x n cubic "board" .
m=int(input('What\'s the biggest board to investigate? '))
for n in range (0, m+1):
    sum=0
    for x1 in range (1, n+1):
      for y1 in range (1, n+1):
        for z1 in range (1, n+1):
          for x2 in range (1, n+1):
            for y2 in range (1, n+1):
              for z2 in range (1, n+1):
                if abs(x1-x2)>1 or abs(y1-y2)>1 or abs(z1-z2)>1:
                  sum=sum+1
    sum=sum//2
    print(n, sum)
(PARI) {a(n) = if(n==0, 0, (n^6 -(3*n-2)^3)/2)}; \\ G. C. Greubel, Apr 03 2019
(Magma) [0] cat [(n^6 -(3*n-2)^3)/2: n in [1..35]]; // G. C. Greubel, Apr 03 2019
(Sage) [0]+[(n^6 -(3*n-2)^3)/2 for n in (1..35)] # G. C. Greubel, Apr 03 2019

Crossrefs

Cf. A061995, A194605.

Sequence in context: A174521 A204711 A125647 * A178265 A201112 A250156

Adjacent sequences: A166537 A166538 A166539 * A166541 A166542 A166543

Keyword

nonn,easy

Author

Michael Engling, Oct 16 2009

Status

approved