A064842 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}.

0, 2, 6, 18, 36, 66, 106, 162, 232, 322, 430, 562, 716, 898, 1106, 1346, 1616, 1922, 2262, 2642, 3060, 3522, 4026, 4578, 5176, 5826, 6526, 7282, 8092, 8962, 9890, 10882, 11936, 13058, 14246, 15506, 16836, 18242, 19722, 21282, 22920, 24642, 26446, 28338, 30316

Offset: 1

Author

N. J. A. Sloane, Oct 25 2001

Keywords

• nonn
• easy

Examples

			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).
		

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).

Crossrefs

Cf. A064843.

Programs

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 *)

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

Extensions

Edited by Emeric Deutsch, Jul 30 2005