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A325603
Lower left-hand y-coordinate for 2 X 2 invisible forest with 0 < x < y.
9
20, 35, 35, 54, 65, 77, 69, 84, 84, 98, 99, 104, 99, 95, 114, 104, 119, 98, 110, 114, 104, 132, 135, 119, 132, 153, 135, 174, 174, 161, 175, 147, 170, 186, 189, 159, 189, 153, 170, 195, 189, 195, 185, 195, 185, 195, 209, 216, 224, 224
OFFSET
1,1
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).
LINKS
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.
EXAMPLE
(14,20), (14,35), (20,35), (44,54), (39,65), (21,77), (45,69), (34,84).
PROG
(SageMath)
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)
KEYWORD
nonn
AUTHOR
Benjamin Hutz, May 10 2019
STATUS
approved