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.

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, 2584, 4181, 6765, 10946, 17711, 3, 72, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 1, 7, 305, 832040, 1346269, 2178309, 3524578, 41, 5702887, 1292, 9227465

Offset: 1

Author

Neal Gersh Tolunsky, Nov 09 2023

Keywords

• nonn

Comments

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.

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.

It appears that the local maxima are the Fibonacci numbers A000045 (except for 1s).

The second-largest values in the log graph, falling roughly on a line, appear to be A001076 (half of the even Fibonacci numbers).

Generalizing the sequence to prohibit the side lengths of any n-gon at distinct n-gonal indices gives A011782.

Examples

			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.
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.
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.
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.
a(7) cannot be 4 because a(4)=2, a(6)=3 and a(7)=4 form a triangle at unsuitable indices.
a(7) can be 5 without contradiction, so a(7)=5.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..2000
• Samuel Harkness, MATLAB program

Crossrefs

Cf. A000045, A001076.

Cf. A229037, A276204, A364057, A361933.

Cf. A316841, A070080 (triangle side lengths).

Programs

MATLAB

See Links.

Extensions

a(11)-a(50) from Samuel Harkness, Nov 13 2023