A245534 a(n) = n^2 + floor(n/2)*(-1)^n.

1, 5, 8, 18, 23, 39, 46, 68, 77, 105, 116, 150, 163, 203, 218, 264, 281, 333, 352, 410, 431, 495, 518, 588, 613, 689, 716, 798, 827, 915, 946, 1040, 1073, 1173, 1208, 1314, 1351, 1463, 1502, 1620, 1661, 1785, 1828, 1958, 2003, 2139, 2186, 2328, 2377, 2525

Offset: 1

Author

Wesley Ivan Hurt, Jul 25 2014

Keywords

• nonn
• easy

Comments

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

Examples

			a(3) = 8; The partitions of 2*3 = 6 into two parts are: (5,1), (4,2), (3,3). Since 3 is odd, we sum the odd numbers from the smallest parts together with the even numbers from the largest parts to get: (1+3) + (4) = 8.
a(4) = 18; 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 even numbers from the smallest parts together with the odd numbers from the largest parts to get: (2+4) + (5+7) = 18.
		

Links

• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A001057, A000290. See A245524 for a very similar sequence.

Programs

Magma

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

Maple

A245534:=n->n^2+floor(n/2)*(-1)^n: seq(A245534(n), n=1..50);

Mathematica

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

PARI

a(n) = n^2 + (n\2)*(-1)^n; \\ Michel Marcus, Aug 06 2014

Formula

G.f.: x*(1 + 4*x + x^2 + 2*x^3)/((1 + x)^2*(1 - x)^3).

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

a(n) = A000290(n) + A001057(n-1) for n > 0.

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) = 23. - Bruno Berselli, May 23 2018