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%I #20 Sep 08 2022 08:46:21
%S 0,5,11,39,60,130,175,306,384,595,715,1025,1196,1624,1855,2420,2720,
%T 3441,3819,4715,5180,6270,6831,8134,8800,10335,11115,12901,13804,
%U 15860,16895,19240,20416,23069,24395,27375,28860,32186,33839,37530,39360,43435,45451
%N a(n) = n*((4*n + 1)*(7*n - 4) + 15*n*(-1)^n)/48.
%C Consider the partitions of n into two parts (p,q). Then 2*a(n) represents the total surface area of the family of rectangular prisms with dimensions p, q and (p + q).
%H Colin Barker, <a href="/A302766/b302766.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(n) = Sum_{i=1..floor(n/2)} n*i + n*(n-i) + i*(n-i).
%F a(n) = floor(n/2)*(6*n^2+3*n-1+3*(n-1)*floor(n/2)-2*floor(n/2)^2)/6.
%F From _Colin Barker_, Apr 13 2018: (Start)
%F G.f.: x^2*(5 + 6*x + 13*x^2 + 3*x^3 + x^4) / ((1 - x)^4*(1 + x)^3).
%F a(n) = (14*n^3 + 3*n^2 - 2*n) / 24 for n even.
%F a(n) = (14*n^3 - 12*n^2 - 2*n) / 24 for n odd.
%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7. (End)
%t Table[Floor[n/2] (6 n^2 + 3 n - 1 + 3 (n - 1) Floor[n/2] - 2 Floor[n/2]^2)/6, {n, 50}]
%o (Magma) [Floor(n/2)*(6*n^2+3*n-1+3*(n-1)*Floor(n/2)-2*Floor(n/2)^2)/6 : n in [1..45]]; // _Vincenzo Librandi_, Apr 13 2018
%o (PARI) a(n) = floor(n/2)*(6*n^2+3*n-1+3*(n-1)*floor(n/2)-2*floor(n/2)^2)/6 \\ _Felix Fröhlich_, Apr 13 2018
%o (PARI) concat(0, Vec(x^2*(5 + 6*x + 13*x^2 + 3*x^3 + x^4) / ((1 - x)^4*(1 + x)^3) + O(x^60))) \\ _Colin Barker_, Apr 13 2018
%o (GAP) List([1..50], n -> n*((4*n+1)*(7*n-4)+15*n*(-1)^n)/48); # _Bruno Berselli_, Apr 16 2018
%Y Cf. A302647, A302758.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Apr 12 2018
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