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A367196 Lexicographically earliest sequence such that for any distinct j, k, m that are the side lengths of a triangle, a(j), a(k), and a(m) are not the side lengths of a triangle.

%I A367196 #33 Nov 24 2023 13:39:56
%S A367196 1,1,1,2,1,3,5,1,8,13,21,2,34,55,89,1,144,233,4,377,610,987,1597,1,17,
%T A367196 2584,4181,6765,10946,17711,3,72,28657,46368,75025,121393,196418,
%U A367196 317811,514229,1,7,305,832040,1346269,2178309,3524578,41,5702887,1292,9227465
%N A367196 Lexicographically earliest sequence such that for any distinct j, k, m that are the side lengths of a triangle, a(j), a(k), and a(m) are not the side lengths of a triangle.
%C A367196 In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z. The empty triangle (or line) is not counted, which means that x + y cannot be equal to z. In practice, if we have two side lengths x and y, we can find their sum s and their difference d, which tells us that side z must fall in the range d < z < s to form a triangle.
%C A367196 For n>0, A002620(n+1) gives the number of combinations of three indices whose corresponding terms cannot be the side lengths of a triangle in this sequence.
%C A367196 It appears that the local maxima are the Fibonacci numbers A000045 (except for 1s).
%C A367196 The second-largest values in the log graph, falling roughly on a line, appear to be A001076 (half of the even Fibonacci numbers).
%C A367196 Generalizing the sequence to prohibit the side lengths of any n-gon at distinct n-gonal indices gives A011782.
%H A367196 Neal Gersh Tolunsky, <a href="/A367196/b367196.txt">Table of n, a(n) for n = 1..2000</a>
%H A367196 Samuel Harkness, <a href="/A367196/a367196_1.m.txt">MATLAB program</a>
%e A367196 a(3)=1 because the indices 1,2,3 could not be the side lengths of a triangle, so there is no restriction and the smallest number is chosen.
%e A367196 a(7) cannot be 1 because a(3)=1, a(5)=1, and a(7)=1 could be the side lengths of a triangle at indices which are also side lengths of a triangle.
%e A367196 a(7) cannot be 2 because a(4)=2, a(6)=3, and a(7)=2 are side lengths of a triangle at indices that forbid it.
%e A367196 a(7) cannot be 3 because a(5)=1, a(6)=3, and a(7)=3 also make a triangle at indices that forbid it.
%e A367196 a(7) cannot be 4 because a(4)=2, a(6)=3 and a(7)=4 form a triangle at unsuitable indices.
%e A367196 a(7) can be 5 without contradiction, so a(7)=5.
%o A367196 (MATLAB) See Links.
%Y A367196 Cf. A000045, A001076.
%Y A367196 Cf. A229037, A276204, A364057, A361933.
%Y A367196 Cf. A316841, A070080 (triangle side lengths).
%K A367196 nonn
%O A367196 1,4
%A A367196 _Neal Gersh Tolunsky_, Nov 09 2023
%E A367196 a(11)-a(50) from _Samuel Harkness_, Nov 13 2023