%I #70 Dec 30 2023 23:44:11
%S 2,3,10,13,28,34,60,70,110,125,182,203,280,308,408,444,570,615,770,
%T 825,1012,1078,1300,1378,1638,1729,2030,2135,2480,2600,2992,3128,3570,
%U 3723,4218,4389,4940,5130,5740,5950,6622,6853,7590,7843,8648,8924,9800,10100,11050,11375
%N a(n) = 1*(n+1-1) + 2*(n+1-2) + ... + k*(n+1-k), where k = floor((n+1)/2).
%C Or, a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n-k+1), where k = floor( n/2 ) and s = (natural numbers).
%C Sum of the areas of the distinct rectangles with positive integer length and width such that L + W = n + 2, W < L. For example, a(5) = 28; the rectangles are 1 X 6, 2 X 5 and 3 X 4. The sum of the areas is then 1*6 + 2*5 + 3*4 = 28. - _Wesley Ivan Hurt_, Nov 12 2017
%H G. C. Greubel, <a href="/A023856/b023856.txt">Table of n, a(n) for n = 1..1000</a>
%H Emanuele Munarini, <a href="https://doi.org/10.4418/2021.76.1.14">Topological indices for the antiregular graphs</a>, Le Mathematiche (2021) Vol. 76, No. 1, see p. 302.
%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) = (n+2)*(4*n^2 + 13*n + 6 - 3(n+2)(-1)^n)/48.
%F a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+2) = -ceiling(n/2)*(ceiling(n/2) + 1)*(2*ceiling(n/2) - 3n - 5)/6. - _Wesley Ivan Hurt_, Sep 20 2013
%F G.f.: x*(2+x+x^2) / ( (1+x)^3*(1-x)^4 ). - _R. J. Mathar_, Sep 25 2013
%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). - _Wesley Ivan Hurt_, Dec 01 2017
%F a(n - 1) = (A000292(n) - (n mod 2) * (ceiling(n / 2)) ^ 2) / 2. - _Luc Rousseau_, Feb 25 2018
%F E.g.f.: (1/24)*( x*(36 + 15*x + 2*x^2)*cosh(x) + (12 + 21*x + 18*x^2 + 2*x^3)*sinh(x) ). - _G. C. Greubel_, Jul 12 2022
%p seq(add(i*(k-i+2), i=1..ceil(k/2)), k=1..70); # _Wesley Ivan Hurt_, Sep 20 2013
%t Table[-Ceiling[n/2]*(Ceiling[n/2]+1)*(2*Ceiling[n/2]-3n-5)/6, {n, 100}] (* _Wesley Ivan Hurt_, Sep 20 2013 *)
%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{2,3,10,13,28,34,60},60] (* _Harvey P. Dale_, Jan 09 2017 *)
%o (Magma) [(n+2)*(4*n^2 + 13*n + 6 - 3*(n+2)*(-1)^n)/48 : n in [1..80]]; // _Wesley Ivan Hurt_, Nov 29 2017
%o (PARI) my(x='x+O('x^99)); Vec(x*(2+x+x^2)/((1+x)^3*(x-1)^4)) \\ _Altug Alkan_, Dec 17 2017
%o (SageMath) [(n+2)*(4*n^2 +13*n +6 -3*(n+2)*(-1)^n)/48 for n in (1..60)] # _G. C. Greubel_, Jul 12 2022
%Y Cf. A000292, A023855, A023857, A024305, A024854.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_
%E Title simplified by _Sean A. Irvine_, Jun 12 2019