OFFSET
1,2
LINKS
G. L. Cohen and E. Tonkes, Dartboard arrangements, Elect. J. Combin., 8(2) (2001), #R4.
Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (2001), 272-273.
Keith Selkirk, Re-designing the dartboard, Math. Gaz., 60 (1976), 171-178.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).
FORMULA
If n mod 2 = 0, then n^3/3 - 4*n/3 + 2 else n^3/3 - 4*n/3 + 1.
a(n) = 2 * A064843(n).
G.f.: -2*x^2*(-1 + x^3 - 2*x^2) / ((1 + x)*(x - 1)^4). - R. J. Mathar, Nov 26 2012
a(n) = (2*n^3 - 8*n + 3*(-1)^n + 9)/6. - Luce ETIENNE, Jul 08 2014
E.g.f.: (2 - x + x^2 + x^3/3)*cosh(x) + (1 - x + x^2 + x^3/3)*sinh(x) - 2. - Stefano Spezia, Apr 13 2024
EXAMPLE
a(4) = 18 because the values of the sum for the permutations of {1, 2, 3, 4} are 10 (8 times), 12 (8 times) and 18 (8 times).
MAPLE
a:=proc(n) if n mod 2 = 0 then (n^3-4*n)/3+2 else (n^3-4*n)/3+1 fi end: seq(a(n), n=1..41); # Emeric Deutsch
MATHEMATICA
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 2, 6, 18, 36}, 45] (* Jean-François Alcover, Apr 01 2020 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 25 2001
EXTENSIONS
Edited by Emeric Deutsch, Jul 30 2005
STATUS
approved