OFFSET
1,1
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).
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Mar 22 2009
EXTENSIONS
More terms from R. J. Mathar, Sep 22 2009
STATUS
approved