A274921 Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.

1, 2, 3, 2, 3, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 1

Offset: 0

Author

Omar E. Pol, Jul 11 2016

Keywords

• nonn

Comments

The structure of the spiral has the following properties:

1) Every 1 is surrounded by three equidistant 2's and three equidistant 3's.

2) Every 2 is surrounded by three equidistant 1's and three equidistant 3's.

3) Every 3 is surrounded by three equidistant 1's and three equidistant 2's.

4) Diagonals are periodic sequences with period 3 (A010882 and A130784).

From Juan Pablo Herrera P., Nov 16 2016: (Start)

5) Every hexagon with a 1 in its center is the same hexagon as the one in the middle of the spiral.

6) Every triangle whose number of numbers is divisible by 3 has the same number of 1's, 2's, and 3's. For example, a triangle with 6 numbers, has two 1's, two 2's, and two 3's. (End)

a(n) = a(n-2) if n > 2 is in A014591, otherwise a(n) = 6 - a(n-1)-a(n-2). - Robert Israel, Sep 15 2017

Examples

			Illustration of initial terms as a spiral:
.
.                3 - 1 - 2 - 3 - 1 - 2
.               /                     \
.              1   2 - 3 - 1 - 2 - 3   1
.             /   /                 \   \
.            2   3   1 - 2 - 3 - 1   2   3
.           /   /   /             \   \   \
.          3   1   2   3 - 1 - 2   3   1   2
.         /   /   /   /         \   \   \   \
.        1   2   3   1   2 - 3   1   2   3   1
.       /   /   /   /   /     \   \   \   \   \
.      2   3   1   2   3   1 - 2   3   1   2   3
.       \   \   \   \   \         /   /   /   /
.        1   2   3   1   2 - 3 - 1   2   3   1
.         \   \   \   \             /   /   /
.          3   1   2   3 - 1 - 2 - 3   1   2
.           \   \   \                 /   /
.            2   3   1 - 2 - 3 - 1 - 2   3
.             \   \                     /
.              1   2 - 3 - 1 - 2 - 3 - 1
.               \
.                3 - 1 - 2 - 3 - 1 - 2
.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001399, A010882, A130784, A253186, A274820, A274821, A274920, A275606, A275610.

Programs

Maple

A[0]:= 1: A[1]:= 2: A[2]:= 3:
b:= 3: c:= 2: d:= 2: e:= 1: f:= 1:
for n from 3 to 200 do
  if n = b then
     r:= b; b:= c + d - f + 1; f:= e; e:= d; d:= c; c:= r;
     A[n]:= A[n-2];
  else
     A[n]:= 6 - A[n-1] - A[n-2];
  fi
od:
seq(A[i],i=0..200); # Robert Israel, Sep 15 2017

Formula

a(n) = A274920(n) + 1.