A152132 Maximal length of rook tour on an n X n+1 board.

2, 8, 24, 54, 104, 174, 270, 396, 558, 756, 996, 1282, 1620, 2010, 2458, 2968, 3546, 4192, 4912, 5710, 6592, 7558, 8614, 9764, 11014, 12364, 13820, 15386, 17068, 18866, 20786, 22832, 25010, 27320, 29768, 32358, 35096, 37982, 41022, 44220, 47582

Offset: 1

Author

R. J. Mathar, Mar 22 2009

Keywords

• nonn
• easy

References

• M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 76.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Crossrefs

Cf. A006071, A087960, A152133, A152134, A152135.

Programs

Magma

I:=[2, 8, 24, 54, 104, 174, 270]; [n le 7 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3) + Self(n-4) - 3*Self(n-5) + 3*Self(n-6)- Self(n-7): n in [1..50]]; // Vincenzo Librandi, Dec 14 2012

Maple

# Figure 43 of the Gardner book:
C := proc(n,m)
if type(m,even) and type(n,even) then
2 ;
elif type(m,odd) and type(n,odd) then
1 ;
elif type(m,even) and type(n,odd) and type(floor(n/2),even) then
3/2 ;
elif type(m,even) and type(n,odd) and type(floor(n/2),odd) then
1/2 ;
elif type(m,odd) and type(n,even) and type(floor(n/2),even) then
0 ;
elif type(m,odd) and type(n,even) and type(floor(n/2),odd) then
1 ;
fi;
end:
# formula for n X m boards, from the Gardner book:
T := proc(n,m)
n*(3*m^2+n^2-10)/6+C(n,m) ;
end:
for n from 1 to 24 do
m := n+3 ; # third diagonal here, for example
printf("%d,",T(n,m)) ;
od:

Mathematica

CoefficientList[Series[-2 * (-1 - x - 2*x^3 - 2*x^4 - 3*x^2 + x^5)/(1 + x)/(x^2 + 1)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 14 2012 *)

Formula

G.f.: -2*x*(-1-x-2*x^3-2*x^4-3*x^2+x^5)/(1+x)/(x^2+1)/(x-1)^4.

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7).

a(n) = 2*n^3/3+n^2-7*n/6+3/4-(-1)^n/4-A087960(n)/2.

Extensions

More terms from R. J. Mathar, Sep 22 2009