A306262 - OEIS

%I #51 May 28 2021 13:12:34

%S 0,0,0,4,11,24,42,68,101,144,196,260,335,424,526,644,777,928,1096,

%T 1284,1491,1720,1970,2244,2541,2864,3212,3588,3991,4424,4886,5380,

%U 5905,6464,7056,7684,8347,9048,9786,10564,11381,12240,13140,14084,15071,16104,17182

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

%H Colin Barker, <a href="/A306262/b306262.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).

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

%F From _Alois P. Heinz_, Feb 01 2019: (Start)

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

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

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

%F 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

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

%p a:= n-> `if`(n=0, 0, (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,

%p     <0|0|0|0|1>, <-1|3|-2|-2|3>>^n. <<1, 0, 0, 4, 11>>)[1, 1]):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 02 2019

%t a[n_] := Module[

%t   {min, max, perm, g, mperm},

%t   perm = Permutations[Range[n]];

%t   g[x_] := Sum[x[[i]] x[[i + 1]], {i, 1, Length[x] - 1}];

%t   mperm = Map[g, perm];

%t   min = Min[mperm];

%t   max = Max[mperm];

%t   Return[max - min]]

%t LinearRecurrence[{3,-2,-2,3,-1},{0,0,0,4,11,24},60] (* _Harvey P. Dale_, Aug 05 2020 *)

%o (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

%Y Cf. A026035, A101986.

%K nonn

%O 0,4

%A _Louis Rogliano_, Feb 01 2019

%E More terms from _Alois P. Heinz_, Feb 01 2019