I am interested in finding out whether the following sequence has already been "discovered" or if it is worth submitting to OEIS (Online Encyclopedia of Integer Sequences), since I can't find it on there or on wolframalpha. Here is the unknown sequence:
1, 4, 15, 56, 214, 833, 3276, 12985, 51720, 206379, 824547, ... [let's call this sequence #1]
This sequence arises when finding area approximations for quarter-circles with radius, r = 1, 2, 4, 8, 16, ..., using squares with side length of 1 unit. Specifically, I am counting how many unit squares fit inside the quarter-circle without crossing the edge of the circle PLUS all the "boundary" squares that cross the edge of the circle.
I found a separate sequence for the number of inside-the-circle unit squares that do not cross the edge of the circle and have identified it as OEIS A156790:
0, 1, 8, 41, 183. 770. 3149. 12730. 51209, 205356, 822500, ... [let's call this sequence #2]
I found another sequence of the number of boundary unit squares that cross the edge of the circle, which is OEIS A000225:
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, ... [let's call this sequence #3]
Adding the respective terms of sequences #2 and #3 together produces sequence #1.
I'm also curious what the explicit and/or recursive formulas would be for each of these sequences.
Edit: I have found the explicit formula for sequence #3 is a(1) = 1, a(n) = 2*(a(n-1)) + 1, for n >= 2.
My goal is to show that as the length of the radius approaches infinity, the average of sequences #1 and #2 divided by the radius-squared approaches pi/4.
Thanks for reading!
