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

Author

Michael Engling, Oct 16 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A061995, A194605.

Programs

Magma

[0] cat [(n^6 -(3*n-2)^3)/2: n in [1..35]]; // 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 *)

PARI

{a(n) = if(n==0,0, (n^6 -(3*n-2)^3)/2)}; \\ G. C. Greubel, Apr 03 2019

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)

Sage

[0]+[(n^6 -(3*n-2)^3)/2 for n in (1..35)] # G. C. Greubel, Apr 03 2019

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