A110610 - OEIS

%I #18 Jan 29 2022 12:43:13

%S 1,4,11,25,48,82,129,191,270,368,487,629,796,990,1213,1467,1754,2076,

%T 2435,2833,3272,3754,4281,4855,5478,6152,6879,7661,8500,9398,10357,

%U 11379,12466,13620,14843,16137,17504,18946,20465,22063,23742,25504

%N 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}.

%H Michael De Vlieger, <a href="/A110610/b110610.txt">Table of n, a(n) for n = 1..10000</a>

%H Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, <a href="https://www.jstor.org/stable/2975240">The Fifty-Seventh William Lowell Putnam Competition</a>, Amer. Math. Monthly, 104, 1997, 744-754, Problem B-3.

%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 (March 2001), pp. 272-273.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F G.f.: x*(1+x)*(1-x+2*x^2-x^3)/(1-x)^4. [_Colin Barker_, Jul 24 2012]

%e 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).

%p 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);

%t 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 *)

%Y Cf. A016825, A110611.

%Y Cf. also A064842, A087035, A185173.

%K nonn,easy

%O 1,2

%A _Emeric Deutsch_, Jul 30 2005