A382518 Let N = A001481(n), the n-th number that is the sum of two nonnegative squares. a(n) is the index of the first lattice-edge sequence that will accept N so that no sequence contains the edges of a triangle, otherwise if no such sequence exists, a(n) = 0.

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 3, 0, 0, 1, 3, 1, 0, 2, 0, 0, 0, 0, 3, 4, 0, 0, 3, 0, 0, 1, 0, 4, 0, 1, 3, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 1, 4

Offset

1,5

Comments

a(n) is only defined where n is the sum of two nonnegative squares. a(n) = 0 is used in all cases where this is untrue.

Conjecture 1: bin #1 contains the orthogonal and 45-degree diagonal lattice edges.

Conjecture 2: After chessboard coloring the lattice, bin #3 contains only lattice edges that connect black and white points.

Links

Table of n, a(n) for n=1..50.

Example

Let's find a(13). a(13) corresponds to the lattice edge connecting {0,0} to {3,2} because 3^2 = 2^2 = 13. to find a(13) we must know all previous values.
a(1), a(2), a(4), a(8) and a(9) are all in bin#1. a(5) and a(10) are both in bin#2. a(13) cannot be in bin#1 because the lattice edges a(1), a(8) and a(13) make a triangle. a(13) cannot be in bin#2 because a(5), a(10) and a(13) form a triangle. a(13) can go into bin#3. a(13) = 3.
Let's find a(32). It goes into bin#1 because no combination of previous lattice edges added to that bin form a triangle that includes the lattice edge corresponding with a(32). a(32) = 1.

Crossrefs

A001481 numbers that are the sum of two nonnegative squares.

A382109 uses the same technique on a cascade of Issai Schur additive sequences.

Sequence in context: A079632 A002654 A113652 * A106139 A350871 A052154

Adjacent sequences: A382515 A382516 A382517 * A382519 A382520 A382521

Keyword

nonn,more

Author

Gordon Hamilton, Mar 29 2025

Status

approved