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A344710
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a(n)/2 is the smallest possible area of a non-obtuse triangle with coordinates in Z^2 and no side shorter than sqrt(n).
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1
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1, 2, 4, 4, 5, 8, 8, 8, 9, 10, 12, 12, 12, 15, 15, 15, 15, 18, 20, 20, 23, 23, 23, 23, 23, 24, 28, 28, 28, 30, 30, 30, 30, 30, 34, 34, 34, 38, 38, 38, 39, 42, 42, 42, 42, 45, 45, 45, 45
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OFFSET
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1,2
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COMMENTS
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The validity of a triangle can be checked via four Diophantine inequalities: three Euclidean norms for the distances, and one derived from the law of cosines 2*max{|AB|,|BC|,|CA|}<=|AB|+|BC|+|CA| for the angles.
If n is not in A001481, a(n) = a(n+1) because a distance of sqrt(n) can never occur in Z^2.
The smallest area of a non-obtuse triangle with an exact rather than a minimum shortest sidelength is documented in A344845. For A344845, only the terms corresponding to A001481/{0} can exist, as only those are norms in Z^2.
For n in A001481, it was proved that a valid triangle with a sidelength of sqrt(2n) or larger has at least area n/2, and that for n in A001481 a valid triangle with area n/2 always exists.
For n in A001481, all valid triangles with an area smaller than n/2 can be found by checking for triangles with no sidelength of sqrt(2n) or longer, and at most one sidelength of sqrt(5n/4) or longer.
These criteria can be used to set A to (0,0), and look for B and C in the set of points X with sqrt(n) <= |AX| < sqrt(5n/4). Furthermore, B can be assumed to be in the first octant, and C in a different octant but at most 2 octants away. Lastly, |AB| <= |AC| can be assumed. It has been shown that for n in A001481, this suffices to find congruent versions of all valid triangles with an area below n/2.
Up to at least n=17, the following sequence [b(n)] has been proved to have the same terms: b(n) = ceiling(1/x(n)), where x(n) is the supremum on the density of marked points ("dots") in the discrete plane Z^2 with pairwise minimum distance sqrt(n).
If it is not identical for larger n, [a(n)] has been shown to at least be an upper bound on [b(n)].
It has been shown that an alternative interpretation of the problem described in b(n) is the packing of circles with diameter sqrt(n) with centers in the discrete plane Z^2.
For [b(n)], it is conjectured that 1/x(n) is always an integer, so the ceiling function can be omitted.
Conjectured next terms (from the conjectured inequality a(n) > sqrt(3/4)*n) are a(50,...,64) ?= 45, 48, 48, 52, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 56.
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LINKS
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FORMULA
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a(n+1) >= a(n).
a(n) > sqrt(3/4)*n (conjectured).
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EXAMPLE
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[a(n)]: For n=4, a triangle with the minimal area of 4/2 = 2 can be placed at A=(0,0), B=(2,0), and C=(0,2). Alternatively, C can be placed at (1,2) or (2,2).
[b(n)]: For n=1, n=2, and n=3, the following repeating patterns (X for dots, O for empty spaces) achieve the highest possible densities of 1, 1/2, and 1/4 respectively:
XXXXXX OXOXOX OXOXOX
XXXXXX XOXOXO OOOOOO
XXXXXX OXOXOX OXOXOX
XXXXXX XOXOXO OOOOOO
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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