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

0, 0, 1, 2, 3, 7, 6, 15, 10, 26, 15, 40, 21, 57, 28, 77, 36, 100, 45, 126, 55, 155, 66, 187, 78, 222, 91, 260, 105, 301, 120, 345, 136, 392, 153, 442, 171, 495, 190, 551, 210, 610, 231, 672, 253, 737, 276, 805, 300, 876, 325, 950, 351, 1027, 378, 1107, 406

Offset: 0

Author

Wesley Ivan Hurt, Jul 23 2014

Keywords

• nonn
• easy

Comments

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).

Union of triangular numbers (A000217) and second pentagonal numbers (A005449). - Wesley Ivan Hurt, Oct 31 2015

Examples

			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.
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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to partitions.
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000217 (triangular numbers), A005449 (second pentagonal numbers).

Cf. A245288.

Programs

Magma

[( 4*n^2-2*n+1-(2*n^2-6*n+1)*(-1)^n )/16 : n in [0..50]];

Maple

A245467:=n->( 4*n^2-2*n+1-(2*n^2-6*n+1)*(-1)^n )/16: seq(A245467(n), n=0..50);

Mathematica

Table[(4n^2 - 2n + 1 - (2n^2 - 6n + 1) (-1)^n)/16, {n, 0, 50}]
CoefficientList[Series[- x^2 (x^3 + 2 x + 1)/((x - 1)^3 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,0,1,2,3,7},60] (* Harvey P. Dale, May 11 2019 *)

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

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

Formula

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.

From Colin Barker, Jul 23 2014: (Start)

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.

G.f.: -x^2*(x^3+2*x+1) / ((x-1)^3*(x+1)^3). (End)

a(2n) = A000217(n), a(2n+1) = A005449(n). - Wesley Ivan Hurt, Oct 31 2015

Sum_{n>=2} 1/a(n) = 8 - Pi/sqrt(3) - 3*log(3). - Amiram Eldar, Aug 25 2022