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A274921
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Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.
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10
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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
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OFFSET
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0,2
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COMMENTS
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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).
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
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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