Formula to Create a Series

Question

Question:

I am looking for a formula to create the series:

1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,...

Steps Tried:

I realized the above series is the sum of two series as shown below.

First Series: 0,2,2,6,6,12,12,20,20,30,30,42,42,56,56,...

Second Series: 1,0,2,0,3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0,...

But I am not sure how to create the above two series. But clearly there is a pattern. The first series is the product of consecutive natural numbers but with one repetition. The second is just natural numbers with zero in between them.

If there is a direct way to create the original series, that might be okay as well. Please let me know.

If this is a very simple question. It has not been simple for me. Hence, please provide steps and helpful suggestions.

Thanks in Advance

Answer 1

Is this $$a_n=\left\lfloor\frac{(n+1)^2}4\right\rfloor?$$

Answer 2

This is OEIS sequence A002620 "Quarter-squares" which has several formulas for the sequence, including $a(n) := \,\textrm{floor}(n^2/4)\,$ for a suitable offset where $\, a(0)=a(1)=0,\,$ $\, a(2)=1.\,$

For your idea of using the sum of two other sequences, let $\, b(n) := \textrm{floor}((n-1)/2)\textrm{floor}((n+1)/2), $ $\, c(n) := (1+(-1)^n)n/2. \,$ Then $\, a(n) = b(n) + c(n). \,$