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A024598 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (odd natural numbers).

%I A024598 #27 Jul 12 2022 15:30:20
%S A024598 1,3,14,22,55,73,140,172,285,335,506,578,819,917,1240,1368,1785,1947,
%T A024598 2470,2670,3311,3553,4324,4612,5525,5863,6930,7322,8555,9005,10416,
%U A024598 10928,12529,13107,14910,15558,17575,18297,20540,21340,23821,24703,27434,28402
%N A024598 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (odd natural numbers).
%C A024598 Sum of the areas of all rectangles with odd side lengths r and s such that r + s = 2n. - _Wesley Ivan Hurt_, Apr 21 2020
%H A024598 Colin Barker, <a href="/A024598/b024598.txt">Table of n, a(n) for n = 1..1000</a>
%H A024598 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A024598 a(n) = (4*n^3 + 3*n^2 + 2*n - 3*n^2*(-1)^n)/12. - _Luce ETIENNE_, Jan 17 2015
%F A024598 G.f.: x*(3*x^4 + 2*x^3 + 8*x^2 + 2*x + 1) / ((x-1)^4*(x+1)^3). - _Colin Barker_, Jan 17 2015
%F A024598 a(n) = Sum_{i=1..n} i * (2*n-i) * (i mod 2). - _Wesley Ivan Hurt_, Apr 21 2020
%F A024598 E.g.f.: (x/6)*( 2*(3 + 3*x + x^2)*cosh(x) + (3 + 9*x + 2*x^2)*sinh(x) ). - _G. C. Greubel_, Jul 12 2022
%t A024598 LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,3,14,22,55,73,140},50] (* _Harvey P. Dale_, Aug 30 2021 *)
%o A024598 (PARI) Vec(x*(3*x^4+2*x^3+8*x^2+2*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Jan 17 2015
%o A024598 (PARI) odd(n) = 2*n-1;
%o A024598 a(n) = sum(j=1, (n+1)\2, odd(j)*odd(n+1-j)); \\ _Michel Marcus_, Jan 17 2015
%o A024598 (Magma) [(4*n^3 +3*n^2 +2*n -3*n^2*(-1)^n)/12: n in [1..50]]; // _G. C. Greubel_, Jul 12 2022
%o A024598 (SageMath) [(4*n^3 +3*n^2 +2*n -3*n^2*(-1)^n)/12 for n in (1..50)] # _G. C. Greubel_, Jul 12 2022
%Y A024598 Cf. A143218.
%K A024598 nonn,easy
%O A024598 1,2
%A A024598 _Clark Kimberling_