A306262 Difference between maximum and minimum sum of products of successive pairs in permutations of [n].

0, 0, 0, 4, 11, 24, 42, 68, 101, 144, 196, 260, 335, 424, 526, 644, 777, 928, 1096, 1284, 1491, 1720, 1970, 2244, 2541, 2864, 3212, 3588, 3991, 4424, 4886, 5380, 5905, 6464, 7056, 7684, 8347, 9048, 9786, 10564, 11381, 12240, 13140, 14084, 15071, 16104, 17182

Offset: 0

Author

Louis Rogliano, Feb 01 2019

Keywords

• nonn

Examples

			a(4) = 11 = 23 - 12. 1342 and 2431 have sums 23, 3214 and 4123 have sums 12.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A026035, A101986.

Programs

Maple

a:= n-> `if`(n=0, 0, (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
    <0|0|0|0|1>, <-1|3|-2|-2|3>>^n. <<1, 0, 0, 4, 11>>)[1, 1]):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 02 2019

Mathematica

a[n_] := Module[
  {min, max, perm, g, mperm},
  perm = Permutations[Range[n]];
  g[x_] := Sum[x[[i]] x[[i + 1]], {i, 1, Length[x] - 1}];
  mperm = Map[g, perm];
  min = Min[mperm];
  max = Max[mperm];
  Return[max - min]]
LinearRecurrence[{3,-2,-2,3,-1},{0,0,0,4,11,24},60] (* Harvey P. Dale, Aug 05 2020 *)

PARI

concat([0,0,0], Vec(x^3*(4 - x - x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Feb 05 2019

Formula

a(n+1) = a(n) + 1/4*((-1+(-1)^(n-1))^2+2*(n-1)*(n+4)) with a(n) = 0 for n <= 2.

From Alois P. Heinz, Feb 01 2019: (Start)

G.f.: -(x^2+x-4)*x^3/((x+1)*(x-1)^4).

a(n) = (2*n^3+6*n^2-26*n+15-3*(-1)^n)/12 for n > 0.

a(n) = A101986(n-1) - A026035(n) for n > 0. (End)

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Wesley Ivan Hurt, May 28 2021

a(n) = A110610(n+1) - A110611(n+1). - Talmon Silver, Sep 24 2025

Extensions

More terms from Alois P. Heinz, Feb 01 2019