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A265645 a(n) = n^2 * floor(n/2).

%I A265645 #57 Mar 15 2024 07:25:21
%S A265645 0,0,4,9,32,50,108,147,256,324,500,605,864,1014,1372,1575,2048,2312,
%T A265645 2916,3249,4000,4410,5324,5819,6912,7500,8788,9477,10976,11774,13500,
%U A265645 14415,16384,17424,19652,20825,23328,24642,27436,28899,32000,33620,37044,38829,42592,44550,48668,50807,55296,57624,62500
%N A265645 a(n) = n^2 * floor(n/2).
%H A265645 Colin Barker, <a href="/A265645/b265645.txt">Table of n, a(n) for n = 0..1000</a>
%H A265645 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A265645 a(n) = A000290(n)*A004526(n). - _Michel Marcus_, Apr 14 2016
%F A265645 G.f.: x^2*(4 + 5*x + 11*x^2 + 3*x^3 + x^4)/((1 - x)^4*(1 + x)^3). - _Ilya Gutkovskiy_, Apr 14 2016; corrected by _Colin Barker_, Apr 14 2016
%F A265645 From _Colin Barker_, Apr 14 2016: (Start)
%F A265645 a(n) = n^2*(2*n + (-1)^n - 1)/4.
%F A265645 a(n) = n^3/2 for n even.
%F A265645 a(n) = n^2*(n-1)/2 for n odd.
%F A265645 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>6. (End)
%F A265645 Sum_{n>=2} 1/a(n) = zeta(3)/4 - Pi^2/4 - 2*log(2) + 4. - _Amiram Eldar_, Mar 15 2024
%t A265645 Table[n^2 Floor[n/2], {n, 0, 50}] (* _Vincenzo Librandi_, Apr 04 2018 *)
%t A265645 LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,0,4,9,32,50,108},60] (* _Harvey P. Dale_, May 19 2019 *)
%o A265645 (Haskell)
%o A265645 seq' x = x^2 * (x `div` 2)
%o A265645 map seq' [0..50]
%o A265645 (PARI) a(n) = n^2*(n\2); \\ _Altug Alkan_, Apr 14 2016
%o A265645 (PARI) concat(vector(2), Vec(x^2*(4+5*x+11*x^2+3*x^3+x^4)/((1-x)^4*(1+x)^3) + O(x^50))) \\ _Colin Barker_, Apr 14 2016
%o A265645 (Magma) [n^2*Floor(n/2): n in [0..50]]; // _Vincenzo Librandi_, Apr 04 2018
%o A265645 (GAP) List([0..55], n -> n^2*Int(n/2)); # _Muniru A Asiru_, Apr 05 2018
%Y A265645 Cf. A000290, A004526.
%K A265645 nonn,easy
%O A265645 0,3
%A A265645 _Ian Stewart_, Apr 06 2016