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

%I A306262 #57 Jul 24 2025 09:55:17
%S A306262 0,0,0,4,11,24,42,68,101,144,196,260,335,424,526,644,777,928,1096,
%T A306262 1284,1491,1720,1970,2244,2541,2864,3212,3588,3991,4424,4886,5380,
%U A306262 5905,6464,7056,7684,8347,9048,9786,10564,11381,12240,13140,14084,15071,16104,17182
%N A306262 Difference between maximum and minimum sum of products of successive pairs in permutations of [n].
%H A306262 Colin Barker, <a href="/A306262/b306262.txt">Table of n, a(n) for n = 0..1000</a>
%H A306262 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A306262 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 A306262 From _Alois P. Heinz_, Feb 01 2019: (Start)
%F A306262 G.f.: -(x^2+x-4)*x^3/((x+1)*(x-1)^4).
%F A306262 a(n) = (2*n^3+6*n^2-26*n+15-3*(-1)^n)/12 for n > 0.
%F A306262 a(n) = A101986(n-1) - A026035(n) for n > 0. (End)
%F A306262 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
%F A306262 a(n) = A110610(n+1) - A110611(n+1). - _Talmon Silver_, Sep 24 2025
%e A306262 a(4) = 11 = 23 - 12. 1342 and 2431 have sums 23, 3214 and 4123 have sums 12.
%p A306262 a:= n-> `if`(n=0, 0, (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
%p A306262     <0|0|0|0|1>, <-1|3|-2|-2|3>>^n. <<1, 0, 0, 4, 11>>)[1, 1]):
%p A306262 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 02 2019
%t A306262 a[n_] := Module[
%t A306262   {min, max, perm, g, mperm},
%t A306262   perm = Permutations[Range[n]];
%t A306262   g[x_] := Sum[x[[i]] x[[i + 1]], {i, 1, Length[x] - 1}];
%t A306262   mperm = Map[g, perm];
%t A306262   min = Min[mperm];
%t A306262   max = Max[mperm];
%t A306262   Return[max - min]]
%t A306262 LinearRecurrence[{3,-2,-2,3,-1},{0,0,0,4,11,24},60] (* _Harvey P. Dale_, Aug 05 2020 *)
%o A306262 (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 A306262 Cf. A026035, A101986.
%K A306262 nonn
%O A306262 0,4
%A A306262 _Louis Rogliano_, Feb 01 2019
%E A306262 More terms from _Alois P. Heinz_, Feb 01 2019