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%I #25 Dec 02 2023 20:36:43
%S 0,0,10,20,60,92,182,248,408,520,770,940,1300,1540,2030,2352,2992,
%T 3408,4218,4740,5740,6380,7590,8360,9800,10712,12402,13468,15428,
%U 16660,18910,20320,22880,24480,27370,29172,32412,34428,38038,40280,44280,46760,51170
%N Sum of the areas of the squares on the sides of the distinct rectangles that can be made with positive integer sides such that L + W = n, W < L.
%H Colin Barker, <a href="/A294473/b294473.txt">Table of n, a(n) for n = 1..1000</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) = 2 * Sum_{i=1..floor((n-1)/2)} i^2 + (n-i)^2.
%F a(n) = 2 * A294286(n).
%F From _Colin Barker_, Oct 31 2017: (Start)
%F G.f.: 2*x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3).
%F a(n) = n*(8*n^2 - 18*n + 4) / 12 for n even.
%F a(n) = n*(8*n^2 - 12*n + 4) / 12 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.
%F (End)
%F a(n) = n*(2*n-3-(-1)^n)*(8*n-3+(-1)^n)/24. - _Wesley Ivan Hurt_, Dec 02 2023
%e a(14) = 1540; the rectangles are 1 X 13, 2 X 12, 3 X 11, 4 X 10, 5 X 9, 6 X 8 (7 X 7 is not considered since W < L). The sum of the areas of the squares on the sides of the rectangles is 2*1^2+2*13^2 + 2*2^2+2*12^2 + 2*3^2+2*11^2 + 2*4^2+2*10^2 + 2*5^2+2*9^2 + 2*6^2+2*8^2 = 340 + 296 + 260 + 232 + 212 + 200 = 1540.
%t Table[2 Sum[i^2 + (n - i)^2, {i, Floor[(n-1)/2]}], {n, 40}]
%o (PARI) concat(vector(2), Vec(2*x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3) + O(x^60))) \\ _Colin Barker_, Oct 31 2017
%o (PARI) a(n) = 2*sum(i=1, (n-1)\2, i^2 + (n-i)^2); \\ _Michel Marcus_, Nov 08 2017
%o (Magma) [n*(2*n-3-(-1)^n)*(8*n-3+(-1)^n)/24: n in [1..60]]; // _Wesley Ivan Hurt_, Dec 02 2023
%Y Cf. A294286, A050187, A294457.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 31 2017
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