OFFSET
1,2
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).
LINKS
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) = 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
EXAMPLE
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.
MATHEMATICA
Table[n^2 + Floor[n/2] (-1)^n, {n, 50}]
PROG
(Magma) [n^2+Floor(n/2)*(-1)^n: n in [1..50]];
(PARI) a(n) = n^2 + (n\2)*(-1)^n; \\ Michel Marcus, Aug 06 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jul 25 2014
STATUS
approved