A110610 Maximal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,...,n}.

1, 4, 11, 25, 48, 82, 129, 191, 270, 368, 487, 629, 796, 990, 1213, 1467, 1754, 2076, 2435, 2833, 3272, 3754, 4281, 4855, 5478, 6152, 6879, 7661, 8500, 9398, 10357, 11379, 12466, 13620, 14843, 16137, 17504, 18946, 20465, 22063, 23742, 25504

Offset

1,2

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, The Fifty-Seventh William Lowell Putnam Competition, Amer. Math. Monthly, 104, 1997, 744-754, Problem B-3.

Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(1)=1; a(n)=(2n^3+3n^2-11n+18)/6 for n>=2.

G.f.: x*(1+x)*(1-x+2*x^2-x^3)/(1-x)^4. [Colin Barker, Jul 24 2012]

Example

a(4)=25 because the values of the sum for the permutations of {1,2,3,4} are 21 (8 times), 24 (8 times) and 25 (8 times).

Maple

a:=proc(n) if n=1 then 1 else (2*n^3+3*n^2-11*n+18)/6 fi end: seq(a(n), n=1..50);

Mathematica

Rest@ CoefficientList[Series[x (1 + x) (1 - x + 2 x^2 - x^3)/(1 - x)^4, {x, 0, 42}], x] (* Michael De Vlieger, Jan 29 2022 *)

Crossrefs

Cf. A016825, A110611.

Cf. also A064842, A087035, A185173.

Sequence in context: A181946 A176959 A115294 * A051462 A006004 A290876

Adjacent sequences: A110607 A110608 A110609 * A110611 A110612 A110613

Keyword

nonn,easy

Author

Emeric Deutsch, Jul 30 2005

Status

approved