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%I #54 Sep 05 2022 12:28:58
%S 0,0,1,1,5,3,12,6,22,10,35,15,51,21,70,28,92,36,117,45,145,55,176,66,
%T 210,78,247,91,287,105,330,120,376,136,425,153,477,171,532,190,590,
%U 210,651,231,715,253,782,276,852,300,925,325,1001,351,1080,378,1162
%N a(n) = (4*n^2 - 2*n - 1 + (2*n^2 - 2*n + 1)*(-1)^n)/16.
%C For even n, the sequence gives the sum of the largest parts of the partitions of n into two parts. For odd n, the sequence gives the sum of the smallest parts of the partitions of n into two parts (see example).
%H Muniru A Asiru, <a href="/A245288/b245288.txt">Table of n, a(n) for n = 0..3000</a> (first 56 terms from Wesley Ivan Hurt)
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F G.f.: x^2*(1+x+2*x^2)/(1-x^2)^3.
%F a(n) = 3*a(n-2)-3*a(n-4)+a(n-6), a(0)=a(1)=0, a(2)=a(3)=1, a(4)=5, a(5)=3.
%F a(n) = (4*n^2 - 2*n - 1 + (2*n^2 - 2*n + 1)*(-1)^n)/16.
%F a(n) = floor(n/2) * ((1+floor(n/2))*(n mod 2) + (2*n-1-floor(n/2))*((n+1) mod 2))/2.
%F Sum_{n>=2} 1/a(n) = 2 - Pi/sqrt(3) + 3*log(3). - _Amiram Eldar_, Aug 25 2022
%F a(2n) = A000326(n), a(2n+1) = A000217(n). - _Wesley Ivan Hurt_, Sep 05 2022
%e a(4) = 5; the partitions of 4 into two parts are: (3,1), (2,2). Since 4 is even, we add the largest parts in these partitions to get 5.
%e a(5) = 3; the partitions of 5 into two parts are: (4,1), (3,2). Since 5 is odd, we add the smallest parts in the partitions to get 3.
%p A245288:=n->(4*n^2-2*n-1+(2*n^2-2*n+1)*(-1)^n)/16: seq(A245288(n),n=0..50);
%t Table[(4n^2 - 2n - 1 + (2n^2 - 2n + 1) (-1)^n)/16, {n, 0, 50}] (* or *)
%t CoefficientList[Series[x^2 (1 + x + 2*x^2)/(1 - x^2)^3, {x, 0, 50}], x]
%o (Magma) [(4*n^2-2*n-1+(2*n^2-2*n+1)*(-1)^n)/16 : n in [0..50]];
%o (PARI) concat([0,0], Vec(x^2*(1+x+2*x^2)/(1-x^2)^3 + O(x^100))) \\ _Colin Barker_, Jul 25 2014
%o (GAP) a := List([0..10^2],n->(4*n^2-2*n-1+(2*n^2-2*n+1)*(-1)^n)/16); # _Muniru A Asiru_, Feb 02 2018
%Y Cf. A245467, A000326 (bisection), A000217 (bisection).
%K nonn,easy
%O 0,5
%A _Wesley Ivan Hurt_, Jul 23 2014
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