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a(n) = n^2 - floor(n/2)*(-1)^n.
3

%I #37 Jun 20 2024 20:42:24

%S 1,3,10,14,27,33,52,60,85,95,126,138,175,189,232,248,297,315,370,390,

%T 451,473,540,564,637,663,742,770,855,885,976,1008,1105,1139,1242,1278,

%U 1387,1425,1540,1580,1701,1743,1870,1914,2047,2093,2232,2280,2425,2475

%N a(n) = n^2 - floor(n/2)*(-1)^n.

%C Consider the partitions of 2n into two parts: When n is odd, a(n) gives the total sum of the even numbers from the smallest parts and the odd numbers from the largest parts of these partitions. When n is even, a(n) gives the total sum of the odd numbers from the smallest parts and the even numbers from the largest parts (see example).

%H Vincenzo Librandi, <a href="/A245524/b245524.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_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = (4*n^2 - 1 - (2*n - 1)*(-1)^n) / 4.

%F a(n) = A000290(n) - A001057(n-1), n > 0.

%F G.f.: x*(1+2*x+5*x^2)/((1+x)^2*(1-x)^3). - _Robert Israel_, Jul 25 2014

%F a(n) = (2*n-1)*(n-floor(n/2)). - _Wesley Ivan Hurt_, Jan 10 2017

%F a(n) = n^2 + Sum_{k=1..n-1} (-1)^k*k for n>1. Example: for n=5, a(5) = 5^2 + (4 - 3 + 2 - 1) = 27. - _Bruno Berselli_, May 23 2018

%F E.g.f.: (1/2)*(x*(2*x+3)*cosh(x) + (2*x^2+x-1)*sinh(x)). - _G. C. Greubel_, Mar 14 2024

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - _Wesley Ivan Hurt_, Jun 20 2024

%e a(3) = 10; The partitions of 2*3 = 6 into two parts are: (5,1), (4,2), (3,3). Since 3 is odd, we sum the even numbers from the smallest parts together with the odd numbers from the largest parts to get: (2) + (3+5) = 10.

%e a(4) = 14; The partitions of 4*2 = 8 into two parts are: (7,1), (6,2), (5,3), (4,4). Since 4 is even, we sum the odd numbers from the smallest parts together with the even numbers from the largest parts to get: (1+3) + (4+6) = 14.

%p A245524:=n->n^2-floor(n/2)*(-1)^n: seq(A245524(n), n=1..50);

%t Table[n^2 - Floor[n/2] (-1)^n, {n, 50}]

%t CoefficientList[Series[(1 + 2 x + 5 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jul 25 2014 *)

%o (Magma) [n^2-Floor(n/2)*(-1)^n : n in [1..50]];

%o (SageMath) [n^2-(-1)^n*(n//2) for n in range(1,71)] # _G. C. Greubel_, Mar 14 2024

%Y Cf. A000290, A001057.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Jul 24 2014