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
Examples
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).
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).
Programs
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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);
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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 *)
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]