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A325603 Lower left-hand y-coordinate for 2 X 2 invisible forest with 0 < x < y.

%I A325603 #26 May 30 2024 15:59:47
%S A325603 20,35,35,54,65,77,69,84,84,98,99,104,99,95,114,104,119,98,110,114,
%T A325603 104,132,135,119,132,153,135,174,174,161,175,147,170,186,189,159,189,
%U A325603 153,170,195,189,195,185,195,185,195,209,216,224,224
%N A325603 Lower left-hand y-coordinate for 2 X 2 invisible forest with 0 < x < y.
%C A325603 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).
%H A325603 Benjamin Hutz, <a href="/A325603/b325603.txt">Table of n, a(n) for n = 1..1000</a>
%H A325603 E. Goins, P. Harris, B. Kubik, A. Mbirika, <a href="https://arxiv.org/abs/1710.04554">Lattice Point Visibility on Generalized Lines of Sight</a>, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
%H A325603 F. Herzog, B. M. Stewart, <a href="https://www.jstor.org/stable/2317753">Patterns of Visible and Nonvisible Lattice Points</a>, Amer. Math. Monthly 78 (1971) 487-496
%H A325603 S. Laishram, F. Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Laishram/lai2.html">Rectangles Of Nonvisible Lattice Points</a>, J. Int. Seq. 18 (2015) 15.10.8.
%e A325603 (14,20), (14,35), (20,35), (44,54), (39,65), (21,77), (45,69), (34,84).
%o A325603 (SageMath)
%o A325603 def is_nxn(x,y,n):
%o A325603     if all([gcd(x+a,y+b) != 1 for a in range(n) for b in range(n)]):
%o A325603         return True
%o A325603     return False
%o A325603 def insert_item(pts, item, index):
%o A325603     N = len(pts)
%o A325603     if N == 0:
%o A325603       return [item]
%o A325603     elif N == 1:
%o A325603         if item[index] < pts[0][index]:
%o A325603             pts.insert(0,item)
%o A325603         else:
%o A325603             pts.append(item)
%o A325603         return pts
%o A325603     else: #binary insertion
%o A325603         left = 1
%o A325603         right = N
%o A325603         mid = ((left + right)/2).floor()
%o A325603         if item[index] < pts[mid][index]:
%o A325603         # item goes into first half
%o A325603             return insert_item(pts[:mid], item, index) + pts[mid:N]
%o A325603         else:
%o A325603         # item goes into second half
%o A325603             return pts[:mid] + insert_item(pts[mid:N], item, index)
%o A325603 B=1200
%o A325603 L=[]
%o A325603 for x in range(1,B):
%o A325603     for y in range(x+1,B):
%o A325603         if is_nxn(x,y,n=2):
%o A325603             G=[x,y,x^2+y^2]
%o A325603             L=insert_item(L, G, 2)
%Y A325603 Cf. A157426, A157427, A157428, A157429, A325602, A325604, A325605, A325606, A325607.
%K A325603 nonn
%O A325603 1,1
%A A325603 _Benjamin Hutz_, May 10 2019