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A325602 Lower left-hand x-coordinate for 2 X 2 invisible forest with 0 < x < y. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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.

EXAMPLE

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

PROG

(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)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Benjamin Hutz, May 10 2019

STATUS

approved

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Last modified March 4 07:25 EST 2021. Contains 341781 sequences. (Running on oeis4.)