Let f = floor and c = ceiling.
a(1) counts this partition: c(1/2) = 1.
a(2) = 5 counts these partitions:
c(5/2) + c(3/2) = 5;
c(5/2) + f(3/2) + c(1/2) = 3 + 1 + 1 = 5.
a(3) = 11 counts these partitions:
c(11/2) + f(6/2) + c(3/2) = 6 + 3 + 2 = 11;
c(11/2) + f(6/2) + f(3/2) + c(1/2) = 6 + 3 + 1 + 1 = 11;
f(11/2) + c(5/2) + c(3/2) + f(2/2) = 5 + 3 + 2 + 1 = 11.
a(4) = 19 counts these partitions:
c(19/2) + f(10/2) + c(5/2) + f(3/2) = 10 + 5 + 3 + 1 = 19;
c(19/2) + f(10/2) + f(5/2) + f(2/2) + c(1/2) = 10 + 5 + 2 + 1 + 1 = 19;
f(19/2) + c(9/2) + c(5/2) + c(3/2) = 9 + 5 + 3 + 2 = 19; f(19/2) + c(9/2) + c(5/2) + f(3/2) + c(1/2) = 9 + 5 + 3 + 1 + 1 = 19.
