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A306262
Difference between maximum and minimum sum of products of successive pairs in permutations of [n].
1
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
0,4
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
EXAMPLE
a(4) = 11 = 23 - 12. 1342 and 2431 have sums 23, 3214 and 4123 have sums 12.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A301045 A008090 A008250 * A099074 A014818 A328684
KEYWORD
nonn
AUTHOR
Louis Rogliano, Feb 01 2019
EXTENSIONS
More terms from Alois P. Heinz, Feb 01 2019
STATUS
approved