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A245467 a(n) = ( 4*n^2 - 2*n + 1 - (2*n^2 - 6*n + 1) * (-1)^n )/16.

%I A245467 #39 Aug 25 2022 04:35:51
%S A245467 0,0,1,2,3,7,6,15,10,26,15,40,21,57,28,77,36,100,45,126,55,155,66,187,
%T A245467 78,222,91,260,105,301,120,345,136,392,153,442,171,495,190,551,210,
%U A245467 610,231,672,253,737,276,805,300,876,325,950,351,1027,378,1107,406
%N A245467 a(n) = ( 4*n^2 - 2*n + 1 - (2*n^2 - 6*n + 1) * (-1)^n )/16.
%C A245467 For even n, the sequence gives the sum of the smallest parts of the partitions of n into two parts. For odd n, the sequence gives the sum of the largest parts of the partitions of n into two parts (see example).
%C A245467 Union of triangular numbers (A000217) and second pentagonal numbers (A005449). - _Wesley Ivan Hurt_, Oct 31 2015
%H A245467 Vincenzo Librandi, <a href="/A245467/b245467.txt">Table of n, a(n) for n = 0..1000</a>
%H A245467 <a href="/index/Par#part">Index entries for sequences related to partitions</a>.
%H A245467 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A245467 a(n) = floor(n/2) * (3*floor(n/2)+1) * (n mod 2)/2 + floor(n/2) * (floor(n/2)+1) * ((n+1) mod 2)/2.
%F A245467 From _Colin Barker_, Jul 23 2014: (Start)
%F A245467 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
%F A245467 G.f.: -x^2*(x^3+2*x+1) / ((x-1)^3*(x+1)^3). (End)
%F A245467 a(2n) = A000217(n), a(2n+1) = A005449(n). - _Wesley Ivan Hurt_, Oct 31 2015
%F A245467 Sum_{n>=2} 1/a(n) = 8 - Pi/sqrt(3) - 3*log(3). - _Amiram Eldar_, Aug 25 2022
%e A245467 a(6) = 6; the partitions of 6 into two parts are: (5,1), (4,2), (3,3). Since 6 is even, we add the smallest parts in these partitions to get 6.
%e A245467 a(7) = 15; the partitions of 7 into two parts are: (6,1), (5,2), (4,3). Since 7 is odd, we add the largest parts in the partitions to get 15.
%p A245467 A245467:=n->( 4*n^2-2*n+1-(2*n^2-6*n+1)*(-1)^n )/16: seq(A245467(n), n=0..50);
%t A245467 Table[(4n^2 - 2n + 1 - (2n^2 - 6n + 1) (-1)^n)/16, {n, 0, 50}]
%t A245467 CoefficientList[Series[- x^2 (x^3 + 2 x + 1)/((x - 1)^3 (x + 1)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jul 25 2014 *)
%t A245467 LinearRecurrence[{0,3,0,-3,0,1},{0,0,1,2,3,7},60] (* _Harvey P. Dale_, May 11 2019 *)
%o A245467 (Magma) [( 4*n^2-2*n+1-(2*n^2-6*n+1)*(-1)^n )/16 : n in [0..50]];
%o A245467 (PARI) concat([0,0], Vec(-x^2*(x^3+2*x+1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ _Colin Barker_, Jul 23 2014
%o A245467 (PARI) vector(100, n, n--; if(n%2==0, t=n/2; t*(t+1)/2, t*(3*t + 1)/2)) \\ _Altug Alkan_, Nov 01 2015
%Y A245467 Cf. A000217 (triangular numbers), A005449 (second pentagonal numbers).
%Y A245467 Cf. A245288.
%K A245467 nonn,easy
%O A245467 0,4
%A A245467 _Wesley Ivan Hurt_, Jul 23 2014