A344845 -id:A344845 - cp's OEIS frontend

A344710 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).

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

Offset: 1

Jonathan F. Waldmann, May 26 2021

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.

			[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

Jonathan F. Waldmann, An algorithm for the upright triangle sequence
Jonathan F. Waldmann, A more nuanced upright triangle sequence
Jonathan F. Waldmann, Proofs for the first few terms in the discrete circle packing sequence
Jonathan F. Waldmann, More proofs for the discrete circle packing sequence

Cf. A001481, A344845.

a(n) = min{A344845(k) | n <= A001481(k+1) < 5n/4}.
a(n+1) >= a(n).
a(n) <= n if n is in A001481.
a(n) > sqrt(3/4)*n (conjectured).

cp's OEIS Frontend

A344710 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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