|
| |
|
|
A361192
|
|
Number of intersections of a grid and (growing) circle with center at a lattice point.
|
|
0
|
|
|
|
1, 4, 12, 8, 12, 20, 12, 20, 16, 20, 28, 20, 28, 20, 28, 36, 28, 36, 32, 36, 28, 36, 28, 44, 36, 44, 36, 44, 40, 44, 36, 44, 52, 44, 52, 44, 52, 44, 52, 44, 52, 60, 48, 60, 52, 60, 52, 60, 52, 60, 52, 60, 68, 52, 68, 60, 68, 64, 68, 60, 68, 60, 68, 60, 68, 76, 68, 76, 60, 76, 68, 76, 68
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Counted intersections are intersections of the circumference of a circle and the grid (all the grid lines together). Beginning with the smallest circle, the radius is increasing, and a new term is added only when the number of intersections changes.
a(n) is a multiple of 4 for all n except 1.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(1)=1 because at the beginning it's just a point. If we start increasing the circle, there would be 4 intersections, so a(2)=4, this holds while the radius is between 0 and 1 (assuming the cells of the grid have side length 1). If the radius is between 1 and sqrt(2), there are 12 intersections, so a(3)=12. After that: r=sqrt(2), a(4)=8; sqrt(2) < r < 2, a(5)=12.
The number of intersections changes when the squared radius reaches a sum of two nonzero squares (A000404) and when it starts exceeding a sum of two squares, so in the latter case there are three consecutive terms of the sequence corresponding to the squared radius smaller than a term of A001481, equal to it, and exceeding it, like a(3)-a(5) in the example above.
|
|
|
MATHEMATICA
|
issq[n_] := n == Floor[Sqrt[n]]^2;
ss[1] = 0; ss[n_] := Product[If[Mod[First@pe, 4] == 1, Last@pe + 1, Boole[EvenQ[Last@pe] || First@pe == 2]], {pe, FactorInteger[n]}] - Boole[issq[n]]; (* A063725, after Charles R Greathouse IV *)
t = 4; a = {1};
Do[AppendTo[a, t - 4 ss[n]]; If[issq[n], t += 8]; AppendTo[a, t], {n, 40}];
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
