A325602 Lower left-hand x-coordinate for 2 X 2 invisible forest with 0 < x < y.

14, 14, 20, 44, 39, 21, 45, 34, 50, 21, 44, 39, 54, 75, 45, 65, 34, 77, 74, 69, 90, 56, 50, 84, 76, 33, 84, 14, 20, 69, 55, 111, 75, 33, 14, 105, 35, 119, 95, 20, 56, 35, 74, 90, 110, 104, 76, 62, 20, 35

Offset: 1

Author

Benjamin Hutz, May 10 2019

Keywords

• nonn

Comments

These are 2 X 2 rectangles of lattice points not visible along straight lines of sight from the origin. The sequence is ordered by Euclidean distance from (0,0).

Examples

			(14,20), (14,35), (20,35), (44,54), (39,65), (21,77), (45,69), (34,84), ...
		

Links

• Benjamin Hutz, Table of n, a(n) for n = 1..1000
• E. Goins, P. Harris, B. Kubik, A. Mbirika, Lattice Point Visibility on Generalized Lines of Sight, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
• F. Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, Amer. Math. Monthly 78 (1971) 487-496
• S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8.

Crossrefs

Cf. A157426, A157427, A157428, A157429.

Cf. A325603, A325604, A325605, A325606, A325607.

Programs

Python

def is_nxn(x,y,n):
    if all([gcd(x+a,y+b) != 1 for a in range(n) for b in range(n)]):
        return True
    return False
def insert_item(pts, item, index):
    N = len(pts)
    if N == 0:
      return [item]
    elif N == 1:
        if item[index] < pts[0][index]:
            pts.insert(0,item)
        else:
            pts.append(item)
        return pts
    else: #binary insertion
        left = 1
        right = N
        mid = ((left + right)/2).floor()
        if item[index] < pts[mid][index]:
        # item goes into first half
            return insert_item(pts[:mid], item, index) + pts[mid:N]
        else:
        # item goes into second half
            return pts[:mid] + insert_item(pts[mid:N], item, index)
B=1200
L=[]
for x in range(1,B):
    for y in range(x+1,B):
        if is_nxn(x,y,n=2):
            G=[x,y,x^2+y^2]
            L=insert_item(L, G, 2)