This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344710 #84 Feb 16 2022 11:56:38
%S A344710 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,
%T A344710 28,28,28,30,30,30,30,30,34,34,34,38,38,38,39,42,42,42,42,45,45,45,45
%N 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).
%C A344710 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.
%C A344710 If n is not in A001481, a(n) = a(n+1) because a distance of sqrt(n) can never occur in Z^2.
%C A344710 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.
%C A344710 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.
%C A344710 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.
%C A344710 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.
%C A344710 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).
%C A344710 If it is not identical for larger n, [a(n)] has been shown to at least be an upper bound on [b(n)].
%C A344710 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.
%C A344710 For [b(n)], it is conjectured that 1/x(n) is always an integer, so the ceiling function can be omitted.
%C A344710 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.
%H A344710 Jonathan F. Waldmann, <a href="https://docs.google.com/spreadsheets/d/19ylTMfMzm9XVvOOYr0xYvFAMOHt0wbQn3x-Q1ObSAIs">An algorithm for the upright triangle sequence</a>
%H A344710 Jonathan F. Waldmann, <a href="https://docs.google.com/spreadsheets/d/19x9Qj8HcmQXVD855IMFH_3MZt70RBwckLld2zYk53L0/edit?usp=sharing">A more nuanced upright triangle sequence</a>
%H A344710 Jonathan F. Waldmann, <a href="https://docs.google.com/spreadsheets/d/1cJo_cP683PiU3ZUKEtNWKvSSMfGMhcI4efwyVwFBb4M">Proofs for the first few terms in the discrete circle packing sequence</a>
%H A344710 Jonathan F. Waldmann, <a href="https://docs.google.com/spreadsheets/d/1bKbl6Qq_PsWtb7zBx_PPmbrlIeVmAMioJ7_gFJRUae0">More proofs for the discrete circle packing sequence</a>
%F A344710 a(n) = min{A344845(k) | n <= A001481(k+1) < 5n/4}.
%F A344710 a(n+1) >= a(n).
%F A344710 a(n) <= n if n is in A001481.
%F A344710 a(n) > sqrt(3/4)*n (conjectured).
%e A344710 [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).
%e A344710 [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:
%e A344710 XXXXXX OXOXOX OXOXOX
%e A344710 XXXXXX XOXOXO OOOOOO
%e A344710 XXXXXX OXOXOX OXOXOX
%e A344710 XXXXXX XOXOXO OOOOOO
%Y A344710 Cf. A001481, A344845.
%K A344710 nonn,more
%O A344710 1,2
%A A344710 _Jonathan F. Waldmann_, May 26 2021