login
A274921
Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.
10
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
0,2
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
LINKS
FORMULA
a(n) = A274920(n) + 1.
EXAMPLE
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
.
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jul 11 2016
STATUS
approved