Finding a pattern, I'm stuck

Question

I have a friend who knows how much I love math, (I imagine new problems just to do the math behind them and to see if I can expand my understanding) and so he brings me the stuff that stumps him. Usually I end up finding the answer for him and then explaining how to him, but he gave me one 2 weeks ago and I cant figure it out. It's part of some math for calculating the area (or maybe perimeter?) of an elipse, the part he wanted me to figure out is: there is a part that goes π * (a + b) * (1 + 1/4 * h + 1/64 * h^2 + 1/256 * h^3 + 1/16384 * h^4 + 1/ 65536 * h^5 + 1/1048576 * h^6 ...) we only have those six, and he wanted me to find the pattern to calculate the rest of the denominators to extend it. But everything I have tried adds extra values within the first six or skips values within the first six. Does anyone know what the pattern is and what formula can be used to calculate the rest?

Answer 1

If I could give some advice, these kinds of questions generally benefit from a look at the Online Encyclopedia of Integer Sequences.

Considering it has to do with ellipses, the sequence of denominators should be A056982.

Answer 2

The $n^\text{th}$ denominator appears to be

$$4^{\large{2n-\text{(the number of 1s in the base-2 representation of }n)}}$$

starting with $n=0.$