Panoramic view of square lattice

[Hugo Pfoertner]

SeqFans,

some days ago I walked over a city square that had a part paved with
quadratic slabs arranged in a simple square lattice. Standing at certain
positions and turning the head shows the lattice points sometimes
arranged such that one can "view" through the gaps and for other view
angles many lattice points are seen in one straight line.

This enticed me to write a small program to find some related sequences:

Number of distinct angular positions under which an observer positioned
in the middle of a square lattice can see the surrounding lattice points
if he makes a 360 degree turn.

Candidate locations for the observer (the origin of the coordinates) are
a lattice point, the center of a square, the center of an edge - see
Table 4.2 p 106 in Sphere Packings, Lattices and Groups.

For the observer at a lattice point one gets the following sequence:

(1/8)*number of distinct angular positions under which an observer
positioned at a lattice point of a square lattice can see the
(2n+1)X(2n+1) points symmetrically surrounding his position.
1,2,4,6,10,12,18,22,28,32,42,46,58,64,72,....
This is http://www.research.att.com/projects/OEIS?Anum=A002088
(sum of totient function) or 2*A046657.

For the observer at a center of a square:

(1/4)*number of distinct angular positions under which an observer
positioned at the center of a square of a square lattice can see the
(2n)X(2n) points symmetrically surrounding his position.

1 3 7 13 19 29 41 49 65 83 95 117 137 155 183 213 233 257 293 317 357
399 423 469 511 543 595 635 671 729 789 825 873 939 983 1053 1125 1165
1225 1303 1357 1439 1503 1559 1647 1719 1779 1851 1947 2007 2107 2209
2257 2363 2471 2543 2655 2743 2815 2911 3021 3101 3201 3327 3411 3541
3649 3721 3857 3995 4087 4207 4319 4403 4551 4701 4797 4917 5073 5177
5309 5471 5551 5717 5873 5981 6153 6273 6389 6567 6747 6867 7011 7171
7279 7469 7661 7757 7953 8151 8283

This sequence is not in OEIS, but superseeker found
a(n+1)-a(n)=phi(2n+1) A037225 (**)

For the observer at the center of an edge:
(1/2)*number of distinct angular positions under which an observer
positioned at the center of an edge of a square lattice can see the
(2n)X(2n-1) points symmetrically surrounding his position.

1 5 13 23 37 55 75 95 127 157 185 227 263 305 357 403 455 511 571 631
703 769 833 923 997 1069 1169 1245 1329 1443 1535 1631 1743 1849 1957
2075 2195 2307 2439 2565 2683 2845 2957 3097 3265 3385 3533 3697 3857
4001 4181 4347 4491 4701 4881 5033 5241 5401 5585 5797 5971 6171 6391
6589 6801 7027 7215 7419 7683 7909 8097 8357 8565 8793 9085 9315 9555
9795 10047 10307 10567 10837 11077 11407 11659 11895 12235 12467 12743
13097 13373 13637 13957 14237 14529 14863 15183 15471 15835 16153 16445

See also http://www.research.att.com/projects/OEIS?Anum=A002939
2*n*(2*n-1).

This is not in OEIS and also superseeker returns [empty]. Can we find
something similar to (**)?

Hugo Pfoertner