(In reply to An alternate approach
by Brian Smith)
The nth term can be expressed as a function by finding the formulas for the two subsequences.
f(n) for odd n = (1/2)*n^2 + (41/2)*n
f(n) for even n = (3/2)*n^2 + (39/2)*n
These two sequences can be combined, utilizing (-1)^n:
f(n) = n^2 + 20*n + ((-1)^n)*((1/2)*n^2-(1/2)*n)
In other words, to calculate the nth term find n^2 + 20*n and the (n-1)th triangular number. If n is even then take the sum and if n is odd then take the difference.