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A325602
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Lower left-hand x-coordinate for 2 X 2 invisible forest with 0 < x < y.
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9
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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
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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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.
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EXAMPLE
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(14,20), (14,35), (20,35), (44,54), (39,65), (21,77), (45,69), (34,84), ...
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PROG
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(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)
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CROSSREFS
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Cf. A157426, A157427, A157428, A157429.
Cf. A325603, A325604, A325605, A325606, A325607.
Sequence in context: A291510 A186128 A105707 * A157427 A003905 A205702
Adjacent sequences: A325599 A325600 A325601 * A325603 A325604 A325605
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KEYWORD
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nonn
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AUTHOR
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Benjamin Hutz, May 10 2019
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STATUS
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approved
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