|
%I #34 Apr 13 2024 16:27:44
%S 0,2,6,18,36,66,106,162,232,322,430,562,716,898,1106,1346,1616,1922,
%T 2262,2642,3060,3522,4026,4578,5176,5826,6526,7282,8092,8962,9890,
%U 10882,11936,13058,14246,15506,16836,18242,19722,21282,22920,24642,26446,28338,30316
%N Maximal value of Sum_{i=1..n} (p(i) - p(i+1))^2, where p(n+1) = p(1), as p ranges over all permutations of {1, 2, ..., n}.
%H G. L. Cohen and E. Tonkes, <a href="https://doi.org/10.37236/1603">Dartboard arrangements</a>, Elect. J. Combin., 8(2) (2001), #R4.
%H Vasile Mihai and Michael Woltermann, <a href="https://www.jstor.org/stable/2695399">Problem 10725: The Smoothest and Roughest Permutations</a>, Amer. Math. Monthly, 108 (2001), 272-273.
%H Keith Selkirk, <a href="https://www.jstor.org/stable/3617473">Re-designing the dartboard</a>, Math. Gaz., 60 (1976), 171-178.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F If n mod 2 = 0, then n^3/3 - 4*n/3 + 2 else n^3/3 - 4*n/3 + 1.
%F a(n) = 2 * A064843(n).
%F G.f.: -2*x^2*(-1 + x^3 - 2*x^2) / ((1 + x)*(x - 1)^4). - _R. J. Mathar_, Nov 26 2012
%F a(n) = (2*n^3 - 8*n + 3*(-1)^n + 9)/6. - _Luce ETIENNE_, Jul 08 2014
%F 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
%e 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).
%p 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_
%t LinearRecurrence[{3, -2, -2, 3, -1}, {0, 2, 6, 18, 36}, 45] (* _Jean-François Alcover_, Apr 01 2020 *)
%Y Cf. A064843.
%K nonn,easy,changed
%O 1,2
%A _N. J. A. Sloane_, Oct 25 2001
%E Edited by _Emeric Deutsch_, Jul 30 2005
|
